{"id":9480,"date":"2025-10-20T18:47:10","date_gmt":"2025-10-20T09:47:10","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9480"},"modified":"2026-09-27T14:32:33","modified_gmt":"2026-09-27T05:32:33","slug":"ch03-limit-of-sequences","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\/","title":{"rendered":"\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/h2>\n\n --><\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">2\uc7a5<\/a>\uc5d0\uc11c \uac70\ub9ac\uacf5\uac04\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uacfc \ucf54\uc2dc \uc870\uac74\uc744 \uc77c\ubc18 \uac70\ub9ac\uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ud3d0\ud3ec, \ucef4\ud329\ud2b8\uc131\ucc98\ub7fc \uac70\ub9ac\ub85c\ubd80\ud130 \uc0dd\uae30\ub294 \uc704\uc0c1\uc801 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\uc758 \uc218\ub834<\/h3>\n<p>\uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc5d0\uc11c \uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \uc810 \\(x\\in X\\)\ub85c <span class=\"defined\">\uc218\ub834<\/span>\ud55c\ub2e4(converge)\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n&gt;N\\)\uc77c \ub54c \\(d(x_n,x)&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc774\uac83\uc744 \\(x_n\\to x\\) \ub610\ub294 \\(\\lim_{n\\to\\infty}x_n=x\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.1. (\uadf9\ud55c\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uc720\uc77c\ud558\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \\(x\\)\uc640 \\(y\\)\uc5d0 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc744 \ud0dd\ud558\uba74<br \/>\n\\[<br \/>\nd(x_n,x)&lt;\\frac{\\varepsilon}{2},\\quad d(x_n,y)&lt;\\frac{\\varepsilon}{2}<br \/>\n\\]<br \/>\n\uac00 \ub3d9\uc2dc\uc5d0 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\nd(x,y)\\leq d(x,x_n)+d(x_n,y)&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc774\ub294 \ubaa8\ub4e0 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c \\(d(x,y)=0\\), \uc989 \\(x=y\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 <span class=\"defined\">\ucf54\uc2dc \uc218\uc5f4<\/span>(Cauchy sequence)\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m&gt;N\\), \\(n&gt;N\\)\uc77c \ub54c \\(d(x_m,x_n)&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \ud56d\uc0c1 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(x_n\\to x\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \\(m,n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd(x_m,x_n)\\leq d(x_m,x)+d(x,x_n)&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\ubaa8\ub4e0 \ucf54\uc2dc \uc218\uc5f4\uc774 \uadf8 \uacf5\uac04\uc758 \uc810\uc73c\ub85c \uc218\ub834\ud558\ub294 \uac70\ub9ac\uacf5\uac04\uc744 <span class=\"defined\">\uc644\ube44\uac70\ub9ac\uacf5\uac04<\/span>(complete metric space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ucf54\uc2dc \uc218\uc5f4\uc774 \ubc18\ub4dc\uc2dc \uc218\ub834\ud55c\ub2e4\ub294 \uc131\uc9c8\uc740 \uac70\ub9ac\uacf5\uac04\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\ud45c\uc900 \uac70\ub9ac \\(d(x,y)=|x-y|\\)\uac00 \uc8fc\uc5b4\uc9c4 \uc720\ub9ac\uc218 \\(\\mathbb Q\\)\ub294 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(\\sqrt2\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc720\ub9ac\uc218\uc5f4\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc \\(\\mathbb Q\\)\uc758 \uc810\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<li>\ud45c\uc900 \uac70\ub9ac\uac00 \uc8fc\uc5b4\uc9c4 \uc5f4\ub9b0\uad6c\uac04 \\((0,1)\\)\uc740 \uc644\ube44\uac00 \uc544\ub2c8\ub2e4. \uc218\uc5f4 \\(\\left\\{\\frac1{n+1}\\right\\}\\)\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc \\((0,1)\\)\uc758 \uc810\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ul>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.2. (\uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc758 \uc644\ube44\uc131)<\/span><\/p>\n<p>\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uac00 \uc8fc\uc5b4\uc9c4 \\(\\mathbb R^d\\)\ub294 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\{x_n\\}\\)\uc774 \\(\\mathbb R^d\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\uace0<br \/>\n\\[<br \/>\nx_n=(x_{n,1},x_{n,2},\\ldots,x_{n,d})<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|x_{m,j}-x_{n,j}|\\leq d_2(x_m,x_n)<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \uac01 \uc88c\ud45c\uc218\uc5f4 \\(\\{x_{n,j}\\}\\)\ub294 \uc2e4\uc218\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4. \\(\\mathbb R\\)\uc758 \uc644\ube44\uc131\uc5d0 \uc758\ud574 \\(x_{n,j}\\to L_j\\)\uc778 \uc2e4\uc218 \\(L_j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(L=(L_1,\\ldots,L_d)\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uac01 \\(j\\)\uc5d0\uc11c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc774\uba74<br \/>\n\\[<br \/>\n|x_{n,j}-L_j|&lt;\\frac{\\varepsilon}{\\sqrt d}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nd_2(x_n,L)<br \/>\n=\\sqrt{\\sum_{j=1}^d|x_{n,j}-L_j|^2}&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(x_n\\to L\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.1.<\/span><br \/>\n\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \\(\\mathbb R^d\\)\uc758 \uc810\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774\uace0, \uac01 \ud56d\uc774<br \/>\n\\[<br \/>\na_n=(a_{n,1},a_{n,2},\\ldots,a_{n,d})<br \/>\n\\]<br \/>\n\uc640 \uac19\uc740 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(L=(L_1,L_2,\\ldots,L_d)\\in\\mathbb R^d\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\{a_n\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(j=1,2,\\ldots,d\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218\uc5f4 \\(\\{a_{n,j}\\}\\)\uac00 \\(L_j\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569<\/h3>\n<p>\\(E\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(x\\in X\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(x\\)\uac00 \\(E\\)\uc758 <span class=\"defined\">\ub0b4\uc810<\/span>(interior point)\uc774\ub77c \ud568\uc740 \\(B(x,r)\\subseteq E\\)\uc778 \uc591\uc218 \\(r\\)\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/li>\n<li>\\(x\\)\uac00 \\(E\\)\uc758 <span class=\"defined\">\uc678\uc810<\/span>(exterior point)\uc774\ub77c \ud568\uc740 \\(B(x,r)\\cap E=\\varnothing\\)\uc778 \uc591\uc218 \\(r\\)\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/li>\n<li>\\(x\\)\uac00 \\(E\\)\uc758 \ub0b4\uc810\ub3c4 \uc544\ub2c8\uace0 \uc678\uc810\ub3c4 \uc544\ub2d0 \ub54c \\(x\\)\ub97c \\(E\\)\uc758 <span class=\"defined\">\uacbd\uacc4\uc810<\/span>(boundary point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(x\\)\uac00 \\(E\\)\uc758 \uacbd\uacc4\uc810\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \uc591\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\cap E\\neq\\varnothing\\)\uc774\uace0 \\(B(x,r)\\cap E^c\\neq\\varnothing\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(E\\)\uc758 \ub0b4\uc810\uc758 \ubaa8\uc784\uc744 \\(E\\)\uc758 <span class=\"defined\">\ub0b4\ubd80<\/span>(interior)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\operatorname{int}(E)\\) \ub610\ub294 \\(E^o\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(E\\)\uc758 \uc678\uc810\uc758 \ubaa8\uc784\uc744 \\(E\\)\uc758 <span class=\"defined\">\uc678\ubd80<\/span>(exterior)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\operatorname{ext}(E)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(E\\)\uc758 \uacbd\uacc4\uc810\uc758 \ubaa8\uc784\uc744 \\(E\\)\uc758 <span class=\"defined\">\uacbd\uacc4<\/span>(boundary)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\operatorname{bd}(E)\\) \ub610\ub294 \\(\\partial E\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<p>\uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(U\\)\uac00 <span class=\"defined\">\uc5f4\ub9b0\uc9d1\ud569<\/span>(open set)\uc774\ub77c\ub294 \uac83\uc740 \\(U\\)\uc758 \ubaa8\ub4e0 \uc810\uc774 \\(U\\)\uc758 \ub0b4\uc810\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(F\\)\uac00 <span class=\"defined\">\ub2eb\ud78c\uc9d1\ud569<\/span>(closed set)\uc774\ub77c\ub294 \uac83\uc740 \uadf8 \uc5ec\uc9d1\ud569 \\(F^c=X\\setminus F\\)\uac00 \\(X\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(\\mathbb R\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569\uc758 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uc5f4\ub9b0\uad6c\uac04 \\((a,b)\\)\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ub2eb\ud78c\uad6c\uac04 \\([a,b]\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ubc18\uc5f4\ub9b0\uad6c\uac04 \\([a,b)\\)\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \ub2eb\ud78c\uc9d1\ud569\ub3c4 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(\\mathbb R\\)\uacfc \\(\\varnothing\\)\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\uba74\uc11c \ub3d9\uc2dc\uc5d0 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.3. (\uc5f4\ub9b0\uc9d1\ud569\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(X\\)\uc640 \\(\\varnothing\\)\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \uc784\uc758\uc758 \ud569\uc9d1\ud569\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \uc720\ud55c \uad50\uc9d1\ud569\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab \ubc88\uc9f8 \uba85\uc81c\ub294 \uc815\uc758\uc5d0\uc11c \ubc14\ub85c \ub530\ub978\ub2e4. \uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784 \\(\\{U_\\alpha\\}_{\\alpha\\in I}\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in\\bigcup_{\\alpha\\in I}U_\\alpha\\)\ub77c\uace0 \ud558\uc790. \uc5b4\ub5a4 \\(\\beta\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\in U_\\beta\\)\uc774\uace0 \\(U_\\beta\\)\uac00 \uc5f4\ub824 \uc788\uc73c\ubbc0\ub85c, \uc5b4\ub5a4 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\subseteq U_\\beta\\subseteq\\bigcup_{\\alpha\\in I}U_\\alpha\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc784\uc758\uc758 \ud569\uc9d1\ud569\ub3c4 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(U_1,\\ldots,U_m\\)\uc774 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(x\\in\\bigcap_{j=1}^mU_j\\)\ub77c\uace0 \ud558\uc790. \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r_j)\\subseteq U_j\\)\uc778 \\(r_j&gt;0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(r=\\min\\{r_1,\\ldots,r_m\\}\\)\ub85c \ub450\uba74<br \/>\n\\[<br \/>\nB(x,r)\\subseteq\\bigcap_{j=1}^mU_j<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \uc720\ud55c \uad50\uc9d1\ud569\ub3c4 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.2.<\/span><br \/>\n\\(A\\)\uc640 \\(B\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A^o\\subseteq A\\)<\/li>\n<li>\\(B\\subseteq B^o\\cup\\partial B\\)<\/li>\n<li>\\(A\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A=A^o\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(B\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\partial B\\subseteq B\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\((A\\cup B)^o\\supseteq A^o\\cup B^o\\)<\/li>\n<li>\\((A\\cap B)^o=A^o\\cap B^o\\)<\/li>\n<li>\\(\\partial(A\\cup B)\\subseteq\\partial A\\cup\\partial B\\)<\/li>\n<li>\\(\\partial(A\\cap B)\\subseteq\\partial A\\cup\\partial B\\)<\/li>\n<\/ol>\n<\/div>\n<p>\\(a\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \uc810\uc774\uace0 \\(U\\)\uac00 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70 \\(a\\in U^o\\)\uc774\uba74 \\(U\\)\ub97c \\(a\\)\uc758 <span class=\"defined\">\uadfc\ubc29<\/span>(neighborhood)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(U\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\uba74 \\(U\\)\ub97c \\(a\\)\uc758 <span class=\"defined\">\uc5f4\ub9b0\uadfc\ubc29<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \ub9cc\uc57d \\(U\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\uba74 \\(U\\)\ub97c \\(a\\)\uc758 <span class=\"defined\">\ub2eb\ud78c\uadfc\ubc29<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \u2018\uc5f4\ub9b0\uadfc\ubc29\u2019\uc744 \u2018\uadfc\ubc29\u2019\uc774\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud55c\ub2e4.]<\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(p(x)\\)\uac00 \ucc38 \ub610\ub294 \uac70\uc9d3\uc73c\ub85c \ud310\ubcc4\ub418\ub294 \uc9c4\uc220\uc77c \ub54c, \u201c\uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(p(x)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\u201d\ub77c\ub294 \ub9d0\uc740 \u201c\uc810 \\(a\\)\uc758 \uadfc\ubc29 \\(U\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in U\\)\uc5d0 \ub300\ud558\uc5ec \\(p(x)\\)\uac00 \ucc38\uc774\ub2e4\u201d\ub77c\ub294 \ub73b\uc774\ub2e4. [\u201c\uc810 \\(a\\)\uc758 \u2018\uc784\uc758\uc758\u2019 \uadfc\ubc29 \\(U\\)\uc5d0\uc11c \\(p(x)\\)\uac00 \ucc38\uc774\ub2e4\u201d\ub77c\ub294 \ub73b\uc774 \uc544\ub2c8\ub2e4.]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.3.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(\\{a_n\\}\\)\uc774 \\(X\\)\uc758 \uc218\uc5f4\uc774\uba70 \\(L\\in X\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\{a_n\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(L\\)\uc758 \uc784\uc758\uc758 \uadfc\ubc29 \\(G\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\notin G\\)\uc778 \ud56d \\(a_n\\)\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc778 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc815\ub9ac 3.3\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2eb\ud78c\uc9d1\ud569\uc758 \uc131\uc9c8\uc744 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.4. (\ub2eb\ud78c\uc9d1\ud569\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(X\\)\uc640 \\(\\varnothing\\)\uc740 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ub2eb\ud78c\uc9d1\ud569\ub4e4\uc758 \uc784\uc758\uc758 \uad50\uc9d1\ud569\uc740 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ub2eb\ud78c\uc9d1\ud569\ub4e4\uc758 \uc720\ud55c \ud569\uc9d1\ud569\uc740 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ub2eb\ud78c\uc9d1\ud569\uc758 \uc5ec\uc9d1\ud569\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac 3.3\uacfc \ub4dc\ubaa8\ub974\uac04 \ubc95\uce59\uc744 \uc801\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\left(\\bigcap_{\\alpha\\in I}F_\\alpha\\right)^c<br \/>\n=\\bigcup_{\\alpha\\in I}F_\\alpha^c,<br \/>\n\\quad<br \/>\n\\left(\\bigcup_{j=1}^mF_j\\right)^c<br \/>\n=\\bigcap_{j=1}^mF_j^c<br \/>\n\\]<br \/>\n\uac00 \uac01\uac01 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \uccab \ubc88\uc9f8 \uc131\uc9c8\ub3c4 \\(X^c=\\varnothing\\), \\(\\varnothing^c=X\\)\uc5d0\uc11c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.4.<\/span><br \/>\n\ub2e4\uc74c \ub450 \uc608\ub97c \ud1b5\ud558\uc5ec \uc815\ub9ac 3.3\uacfc \uc815\ub9ac 3.4\uc5d0\uc11c \u2018\uc720\ud55c\u2019\uc774\ub77c\ub294 \uc870\uac74\uc744 \uc77c\ubc18\uc801\uc73c\ub85c \uc81c\uac70\ud560 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0 \uad50\uc9d1\ud569\uc774 \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2d0 \uc218 \uc788\ub294 \uc608\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<li>\ub2eb\ud78c\uc9d1\ud569\ub4e4\uc758 \uac00\uc0b0 \ud569\uc9d1\ud569\uc774 \ub2eb\ud78c\uc9d1\ud569\uc774 \uc544\ub2d0 \uc218 \uc788\ub294 \uc608\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\ubd80\ubd84\uacf5\uac04\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.5. (\ubd80\ubd84\uacf5\uac04\uc5d0\uc11c\uc758 \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569)<\/span><\/p>\n<p>\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(Y\\)\uac00 \\(X\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\uba70 \\(E\\subseteq Y\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(E\\)\uac00 \\(Y\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(X\\)\uc5d0\uc11c\uc758 \uc5f4\ub9b0\uc9d1\ud569 \\(G\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(E=Y\\cap G\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(E\\)\uac00 \\(Y\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(X\\)\uc5d0\uc11c\uc758 \ub2eb\ud78c\uc9d1\ud569 \\(F\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(E=Y\\cap F\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(E\\)\uac00 \\(Y\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(y\\in E\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(r_y&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nB_Y(y,r_y)=Y\\cap B_X(y,r_y)\\subseteq E<br \/>\n\\]<br \/>\n\uc774\ub2e4. \\(G=\\bigcup_{y\\in E}B_X(y,r_y)\\)\ub85c \ub450\uba74 \\(G\\)\ub294 \\(X\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(E=Y\\cap G\\)\uc774\ub2e4. \uc5ed\uc73c\ub85c \\(E=Y\\cap G\\)\uc774\uace0 \\(G\\)\uac00 \\(X\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(y\\in E\\)\uc774\uba74 \uc5b4\ub5a4 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B_X(y,r)\\subseteq G\\)\uc774\uace0, \ub530\ub77c\uc11c \\(B_Y(y,r)\\subseteq E\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\ub294 \\(Y\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\ub450 \ubc88\uc9f8 \uba85\uc81c\ub294 \uccab \ubc88\uc9f8 \uba85\uc81c\uc640 \uc5ec\uc9d1\ud569\uc744 \uc774\uc6a9\ud558\uba74 \ub41c\ub2e4. \uc2e4\uc81c\ub85c \\(E\\)\uac00 \\(Y\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(Y\\setminus E\\)\uac00 \\(Y\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc778 \uac83\uc774\uace0, \uc774\ub294 \uc5b4\ub5a4 \uc5f4\ub9b0\uc9d1\ud569 \\(G\\subseteq X\\)\uc5d0 \ub300\ud558\uc5ec \\(Y\\setminus E=Y\\cap G\\)\uc778 \uac83\uacfc \ub3d9\uce58\uc774\ub2e4. \uc774\ub54c \\(F=X\\setminus G\\)\ub85c \ub450\uba74 \\(F\\)\ub294 \\(X\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc774\uace0 \\(E=Y\\cap F\\)\uc774\ub2e4. \uc5ed\ub3c4 \uac19\uc740 \uacc4\uc0b0\uc73c\ub85c \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.5.<\/span><br \/>\n\\(Y=[0,1)\\)\uc744 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc73c\ub85c \uc0dd\uac01\ud558\uc790. \\([0,1\/2)\\)\ub294 \\(Y\\)\uc5d0\uc11c\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774\uc9c0\ub9cc \\(\\mathbb R\\)\uc5d0\uc11c\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774 \uc544\ub2c8\uba70, \\([1\/2,1)\\)\uc740 \\(Y\\)\uc5d0\uc11c\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\uc9c0\ub9cc \\(\\mathbb R\\)\uc5d0\uc11c\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\\(A\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(A\\)\ub97c \ud3ec\ud568\ud558\ub294 \ubaa8\ub4e0 \ub2eb\ud78c\uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc744 \\(A\\)\uc758 <span class=\"defined\">\ud3d0\ud3ec<\/span>(closure) \ub610\ub294 <span class=\"defined\">\ub2eb\uac1c<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\overline A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc810 \\(x\\in X\\)\uac00 \uc9d1\ud569 \\(A\\)\uc758 <span class=\"defined\">\uc9d1\uc801\uc810<\/span>(cluster point)\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(r&gt;0\\)\uc5d0 \ub300\ud574 \\(B'(x,r)\\cap A\\neq\\varnothing\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \\(A\\)\uc758 \uc9d1\uc801\uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(A\\)\uc758 <span class=\"defined\">\ub3c4\uc9d1\ud569<\/span>(derived set)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(A&#8217;\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 3.6. (\ud3d0\ud3ec\uc758 \ud2b9\uc131)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\)\uc640 \uc810 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(x\\in\\overline A\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\cap A\\neq\\varnothing\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\n\\[<br \/>\n\\overline A=A\\cup A&#8217;.\\tag{3.1}<br \/>\n\\]\n<\/li>\n<li>\\(x\\in\\overline A\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\)\uc758 \uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc5b4\ub5a4 \uc218\uc5f4\uc774 \\(x\\)\ub85c \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(x\\notin\\overline A\\)\ub77c\uace0 \ud558\uc790. \\(\\overline A\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \\(X\\setminus\\overline A\\)\ub294 \\(x\\)\uc758 \uc5f4\ub9b0\uadfc\ubc29\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\cap A=\\varnothing\\)\uc774\ub2e4. \uc5ed\uc73c\ub85c \\(B(x,r)\\cap A=\\varnothing\\)\uc778 \\(r&gt;0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\ubb38\uc81c 2.2<\/a>\uc5d0\uc11c \ud655\uc778\ud55c \uac83\ucc98\ub7fc \uc5f4\ub9b0\uacf5\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\ubbc0\ub85c \\(X\\setminus B(x,r)\\)\uc740 \\(A\\)\ub97c \ud3ec\ud568\ud558\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \\(x\\notin\\overline A\\)\uc774\ub2e4. \uc774\ub85c\uc368 \uccab \ubc88\uc9f8 \uba85\uc81c\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uccab \ubc88\uc9f8 \uba85\uc81c\uc5d0 \uc758\ud574 \\(x\\in A\\)\uc774\uba74 \\(x\\in\\overline A\\)\uc774\uace0, \\(x\\notin A\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(x\\in\\overline A\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \ubaa8\ub4e0 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B'(x,r)\\cap A\\neq\\varnothing\\)\uc778 \uac83\uc774\ub2e4. \ub530\ub77c\uc11c \ub450 \ubc88\uc9f8 \uba85\uc81c\uac00 \ub530\ub978\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(x\\in\\overline A\\)\ub77c\uace0 \ud558\uc790. \\(x\\in A\\)\uc774\uba74 \uc0c1\uc218\uc218\uc5f4 \\(a_n=x\\)\ub97c \ud0dd\ud558\uba74 \ub41c\ub2e4. \\(x\\notin A\\)\uc774\uba74 \uac01 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n\\in A\\cap B\\left(x,\\frac1n\\right)<br \/>\n\\]<br \/>\n\uc778 \uc810\uc744 \ud558\ub098\uc529 \ud0dd\ud560 \uc218 \uc788\uace0, \uadf8\ub7ec\uba74 \\(a_n\\to x\\)\uc774\ub2e4. \uc5ed\uc73c\ub85c \\(a_n\\in A\\)\uc774\uace0 \\(a_n\\to x\\)\uc774\uba74 \uc784\uc758\uc758 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(a_n\\in B(x,r)\\cap A\\)\uc774\ubbc0\ub85c \uccab \ubc88\uc9f8 \uba85\uc81c\uc5d0 \uc758\ud574 \\(x\\in\\overline A\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc911\uc694\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.7. (\ub2eb\ud78c\uc9d1\ud569\uc758 \ud2b9\uc131)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(F\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc740 \ubaa8\ub450 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc9d1\ud569 \\(F\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(F=\\overline F\\)<\/li>\n<li>\\(F&#8217;\\subseteq F\\)<\/li>\n<li>\\(F\\)\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774 \uc218\ub834\ud55c\ub2e4\uba74 \uadf8 \uadf9\ud55c\uc774 \ubc18\ub4dc\uc2dc \\(F\\)\uc5d0 \uc18d\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(F\\subseteq\\overline F\\)\ub294 \ud56d\uc0c1 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \\(F\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\uba74 \\(\\overline F\\subseteq F\\)\uc774\uace0 \\(F=\\overline F\\)\uc774\ub2e4. \ubc18\ub300\ub85c \\(F=\\overline F\\)\uc774\uba74 \\(\\overline F\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \\(F\\)\ub3c4 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4. \uc815\ub9ac 3.6\uc758 \ub4f1\uc2dd \\(\\overline F=F\\cup F&#8217;\\)\uc5d0 \uc758\ud574 \\(F=\\overline F\\)\uc640 \\(F&#8217;\\subseteq F\\)\ub294 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(F\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\uace0 \\(x_n\\in F\\), \\(x_n\\to x\\)\ub77c\uace0 \ud558\uc790. \uc815\ub9ac 3.6\uc5d0 \uc758\ud574 \\(x\\in\\overline F=F\\)\uc774\ub2e4. \uc5ed\uc73c\ub85c \uc218\uc5f4\uc5d0 \uad00\ud55c \uc870\uac74\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(x\\in\\overline F\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ub9ac 3.6\uc5d0 \uc758\ud574 \\(x_n\\in F\\)\uc774\uace0 \\(x_n\\to x\\)\uc778 \uc218\uc5f4\uc774 \uc874\uc7ac\ud558\ubbc0\ub85c \\(x\\in F\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\overline F\\subseteq F\\)\uc774\uace0 \\(F\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.6.<\/span><br \/>\n\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\partial A=\\overline A\\setminus A^o,<br \/>\n\\quad<br \/>\n\\overline A=A^o\\cup\\partial A<br \/>\n\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.7.<\/span><br \/>\n\\(A=\\{1\/n\\mid n\\in\\mathbb N\\}\\subseteq\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \\(A&#8217;\\), \\(\\overline A\\), \\(A^o\\), \\(\\partial A\\)\ub97c \uac01\uac01 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.8.<\/span><br \/>\n\\(E\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ub2eb\ud78c\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \\(x\\in\\overline E\\setminus E\\)\ub97c \ud558\ub098 \ud0dd\ud558\uc5ec, \ubaa8\ub4e0 \ud56d\uc774 \\(E\\)\uc5d0 \uc18d\ud558\uba74\uc11c \\(x\\)\ub85c \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc744 \uc9c1\uc811 \uad6c\uc131\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.9.<\/span><br \/>\n\\(X\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(E\\)\uac00 \\(X\\)\uc758 \ubd80\ubd84\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(E\\)\uac00 \uc644\ube44\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(E\\)\uac00 \\(X\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\)\uac00 <span class=\"defined\">\uc870\ubc00<\/span>(dense)\ud558\ub2e4\ub294 \uac83\uc740 \\(\\overline A=X\\)\uc778 \uac83\uc774\ub2e4. \uc870\ubc00\ud55c \uac00\uc0b0\ubd80\ubd84\uc9d1\ud569\uc744 \uac00\uc9c4 \uacf5\uac04\uc744 <span class=\"defined\">\uac00\ubd84\uacf5\uac04<\/span>(separable space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ul>\n<li>\\(\\mathbb Q\\)\ub294 \\(\\mathbb R\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4.<\/li>\n<li>\\(\\mathbb R^n\\)\uc740 \uac00\ubd84\uacf5\uac04\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(\\mathbb Q^n\\)\uc740 \uac00\uc0b0\uc9d1\ud569\uc774\uace0 \\(\\mathbb R^n\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4.<\/li>\n<li>\ube44\uac00\uc0b0 \uac1c\uc758 \uc810\uc744 \uac00\uc9c4 \uc774\uc0b0\uac70\ub9ac\uacf5\uac04\uc740 \uac00\ubd84\uacf5\uac04\uc774 \uc544\ub2c8\ub2e4. \uc774\uc0b0\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \ud55c\uc6d0\uc18c\uc9d1\ud569\uc774 \uc5f4\ub9b0\uc9d1\ud569\uc774\ubbc0\ub85c \uc870\ubc00\ud55c \ubd80\ubd84\uc9d1\ud569\uc740 \uacf5\uac04 \uc804\uccb4\uc640 \uac19\uc544\uc57c \ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>\ubd80\ubd84\uc218\uc5f4\uc758 \uadf9\ud55c<\/h3>\n<p>\uc218\uc5f4 \\(\\{x_n\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \\(\\{x_{n_k}\\}\\)\uac00 \\(x\\)\ub85c \uc218\ub834\ud560 \ub54c, \\(x\\)\ub97c \\(\\{x_n\\}\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc218\uc5f4\uadf9\ud55c<\/span>(subsequential limit)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9d1\ud569\uc758 \uc9d1\uc801\uc810\uacfc \ud63c\ub3d9\ud558\uc9c0 \uc54a\uae30 \uc704\ud558\uc5ec \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc218\uc5f4\uc5d0 \ub300\ud574\uc11c\ub294 \u2018\ubd80\ubd84\uc218\uc5f4\uadf9\ud55c\u2019\uc774\ub77c\ub294 \uc6a9\uc5b4\ub97c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.10.<\/span><br \/>\n\uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \\(L\\)\ub85c \uc218\ub834\ud558\uace0 \\(\\{x_{n_k}\\}\\)\uac00 \\(\\{x_n\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\uba74, \\(\\{x_{n_k}\\}\\)\ub3c4 \\(L\\)\ub85c \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc720\uacc4\uc778 \uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\ns_n=\\sup\\{a_k\\mid k\\geq n\\},\\quad<br \/>\n\\ell_n=\\inf\\{a_k\\mid k\\geq n\\}<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \\(\\{s_n\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba70, \\(\\{\\ell_n\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \ub450 \uc218\uc5f4\uc740 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uc774\ub54c <span class=\"defined\">\uc0c1\uadf9\ud55c<\/span>(limit superior)\uacfc <span class=\"defined\">\ud558\uadf9\ud55c<\/span>(limit inferior)\uc744<br \/>\n\\[<br \/>\n\\varlimsup_{n\\to\\infty}a_n<br \/>\n=\\lim_{n\\to\\infty}\\left(\\sup\\{a_k\\mid k\\geq n\\}\\right),\\quad<br \/>\n\\varliminf_{n\\to\\infty}a_n<br \/>\n=\\lim_{n\\to\\infty}\\left(\\inf\\{a_k\\mid k\\geq n\\}\\right).\\tag{3.2}<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\ell_n\\leq s_n\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\varliminf_{n\\to\\infty}a_n\\leq\\varlimsup_{n\\to\\infty}a_n.\\tag{3.3}<br \/>\n\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc720\uacc4\uac00 \uc544\ub2cc \uc218\uc5f4\uc5d0 \ub300\ud574\uc11c\ub3c4 \ud655\uc7a5\uc2e4\uc218\uac12 \\(\\pm\\infty\\)\ub97c \ud5c8\uc6a9\ud558\uc5ec \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc744 \uc815\uc758\ud560 \uc218 \uc788\uc9c0\ub9cc, \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc720\uacc4\uc778 \uc218\uc5f4\ub9cc \ub2e4\ub8ec\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.11.<\/span><br \/>\n\uc74c\uc774 \uc544\ub2cc \uc720\uacc4 \uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\), \\(\\{b_n\\}\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n\\[<br \/>\n\\varlimsup_{n\\to\\infty}(a_nb_n)<br \/>\n\\leq<br \/>\n\\varlimsup_{n\\to\\infty}a_n\\cdot\\varlimsup_{n\\to\\infty}b_n.<br \/>\n\\]<br \/>\n\ub610\ud55c \ub4f1\ud638\uac00 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294 \uc608\ub97c \ucc3e\uc73c\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 3.8. (\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc758 \uae30\ubcf8\uc131\uc9c8)<\/span><\/p>\n<p>\uc720\uacc4\uc778 \uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc740 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc0c1\uadf9\ud55c\uc740 \ubd80\ubd84\uc218\uc5f4\uadf9\ud55c \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc774\uace0, \ud558\uadf9\ud55c\uc740 \ubd80\ubd84\uc218\uc5f4\uadf9\ud55c \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac12\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc774 \uc77c\uce58\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab \ubc88\uc9f8 \uba85\uc81c\ub294 \uc815\uc758 \uc9c1\uc804\uc5d0 \ubcf4\uc778 \ub2e8\uc870\uc131 \ubc0f \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0\uc11c \ub530\ub978\ub2e4. \\(L=\\varlimsup a_n\\)\uc774\ub77c\uace0 \ud558\uc790. \\(s_n=\\sup\\{a_k\\mid k\\geq n\\}\\)\uc774\uba74 \\(s_n\\downarrow L\\)\uc774\ub2e4. \\(n_0=0\\)\uc73c\ub85c \ub450\uace0 \uadc0\ub0a9\uc801\uc73c\ub85c \\(N_j&gt;n_{j-1}\\)\uc774\uace0 \\(s_{N_j}&lt;L+1\/j\\)\uac00 \ub418\ub3c4\ub85d \\(N_j\\)\ub97c \ud0dd\ud55c \ub4a4, \uc0c1\ud55c\uc758 \uc815\uc758\uc5d0 \ub530\ub77c \\(n_j\\geq N_j\\)\uc774\uba74\uc11c<br \/>\n\\[<br \/>\na_{n_j}&gt;s_{N_j}-\\frac1j<br \/>\n\\]<br \/>\n\uac00 \ub418\ub3c4\ub85d \ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\nL-\\frac1j\\leq s_{N_j}-\\frac1j&lt;a_{n_j}\\leq s_{N_j}&lt;L+\\frac1j<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(a_{n_j}\\to L\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(L\\)\uc740 \ubd80\ubd84\uc218\uc5f4\uadf9\ud55c\uc774\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \\(a_{n_j}\\to M\\)\uc778 \uc784\uc758\uc758 \ubd80\ubd84\uc218\uc5f4\uc744 \ud0dd\ud558\uc790. \uc784\uc758\uc758 \\(N\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(j\\)\uc774\uba74 \\(n_j\\geq N\\)\uc774\ubbc0\ub85c \\(a_{n_j}\\leq s_N\\)\uc774\ub2e4. \\(j\\to\\infty\\)\ub85c \ubcf4\ub0b4\uba74 \\(M\\leq s_N\\)\uc774\uace0, \ub2e4\uc2dc \\(N\\to\\infty\\)\ub85c \ubcf4\ub0b4\uba74 \\(M\\leq L\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(L\\)\uc740 \ubd80\ubd84\uc218\uc5f4\uadf9\ud55c \uc911 \ucd5c\ub313\uac12\uc774\ub2e4. \ud558\uadf9\ud55c\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac19\uc740 \ub17c\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \ucd5c\uc19f\uac12\uc784\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(a_n\\to A\\)\uc774\uba74 \ubaa8\ub4e0 \ubd80\ubd84\uc218\uc5f4\ub3c4 \\(A\\)\ub85c \uc218\ub834\ud558\ubbc0\ub85c \ub450 \ubc88\uc9f8 \uba85\uc81c\uc5d0 \uc758\ud574 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc774 \ubaa8\ub450 \\(A\\)\uc774\ub2e4. \uc5ed\uc73c\ub85c<br \/>\n\\[<br \/>\n\\varliminf a_n=\\varlimsup a_n=L<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \\(\\ell_n\\leq a_n\\leq s_n\\)\uc774\uace0 \\(\\ell_n\\to L\\), \\(s_n\\to L\\)\uc774\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(a_n\\to L\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc704 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uba74 \uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc744 \ube44\uad50\uc801 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(a_n=(-1)^n\\left(1+\\frac1n\\right)\\)\uc73c\ub85c \uc815\uc758\ub41c \uc218\uc5f4 \\(\\{a_n\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uadf9\ud55c\uc740 \\(-1\\)\uacfc \\(1\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\varlimsup_{n\\to\\infty}a_n=1,\\quad<br \/>\n\\varliminf_{n\\to\\infty}a_n=-1<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.12.<\/span><br \/>\n\uc720\uacc4\uc778 \uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\)\uacfc \uc2e4\uc218 \\(L\\)\uc5d0 \ub300\ud558\uc5ec \\(L=\\varlimsup a_n\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \ub2e4\uc74c \ub450 \uc870\uac74\uc778 \uac83\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc774\uba74 \\(a_n&lt;L+\\varepsilon\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n&gt;L-\\varepsilon\\)\uc778 \\(n\\)\uc774 \ubb34\ud55c\ud788 \ub9ce\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\uc5f4\uc758 \uac01 \ud56d\uc774 \ub2e4\uc2dc \uc218\uc5f4\uc778 \uc0c1\ud669\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \uc88c\ud45c\uc5d0\uc11c \ub3d9\uc2dc\uc5d0 \uc218\ub834\ud558\ub294 \ud558\ub098\uc758 \ubd80\ubd84\uc218\uc5f4\uc744 \uace8\ub77c\uc57c \ud560 \ub54c\uac00 \uc788\ub2e4. \ub2e4\uc74c \uc815\ub9ac\ub294 \uc774\ub97c \uc704\ud55c <span class=\"defined\">\ub300\uac01\uc120 \ub17c\ubc95<\/span>(diagonal argument)\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.9. (\ub300\uac01\uc120 \ub17c\ubc95)<\/span><\/p>\n<p>\uac01 \uc591\uc758 \uc815\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4 \\(\\{a_n^{(k)}\\}_{n=1}^{\\infty}\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc99d\uac00\ud558\ub294 \uc790\uc5f0\uc218\uc5f4<br \/>\n\\[<br \/>\nn_1&lt;n_2&lt;n_3&lt;\\cdots<br \/>\n\\]<br \/>\n\uc774 \uc874\uc7ac\ud558\uc5ec, \ubaa8\ub4e0 \uace0\uc815\ub41c \\(k\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ubd84\uc218\uc5f4 \\(\\{a_{n_j}^{(k)}\\}_{j=1}^{\\infty}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac\uc5d0 \uc758\ud574 \uccab \ubc88\uc9f8 \uc218\uc5f4 \\(\\{a_n^{(1)}\\}\\)\uc5d0\uc11c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uace8\ub77c \uadf8 \ucca8\uc790\uc5f4\uc744<br \/>\n\\[<br \/>\nn_1^{(1)}&lt;n_2^{(1)}&lt;\\cdots<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774 \ucca8\uc790\uc5f4\uc744 \ub530\ub77c \ub450 \ubc88\uc9f8 \uc218\uc5f4\uc744 \ubcf4\uba74 \uc5ec\uc804\ud788 \uc720\uacc4\uc774\ubbc0\ub85c, \ub2e4\uc2dc \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uace8\ub77c<br \/>\n\\[<br \/>\nn_1^{(2)}&lt;n_2^{(2)}&lt;\\cdots<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \ub450 \ubc88\uc9f8 \ucca8\uc790\uc5f4\uc740 \uccab \ubc88\uc9f8 \ucca8\uc790\uc5f4\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec, \uac01 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\{n_j^{(k+1)}\\}\\)\uac00 \\(\\{n_j^{(k)}\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\uace0 \\(\\{a_{n_j^{(k)}}^{(k)}\\}\\)\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\uc81c \\(n_j=n_j^{(j)}\\)\ub85c \ub454\ub2e4. \uac01 \ub2e8\uacc4\uc758 \ucca8\uc790\uc5f4\uc774 \uc55e \ub2e8\uacc4 \ucca8\uc790\uc5f4\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nn_j^{(j)}\\geq n_j^{(j-1)}&gt;n_{j-1}^{(j-1)},<br \/>\n\\]<br \/>\n\uc989 \\(n_1&lt;n_2&lt;\\cdots\\)\uc774\ub2e4. \uace0\uc815\ub41c \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(j\\geq k\\)\uc774\uba74 \\(n_j\\)\ub294 \\(k\\)\ubc88\uc9f8 \ub2e8\uacc4\uc5d0\uc11c \uc5bb\uc740 \ucca8\uc790\uc5f4\uc758 \uc6d0\uc18c\uc774\ubbc0\ub85c, \\(\\{a_{n_j}^{(k)}\\}_{j\\geq k}\\)\ub294 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\{a_{n_j^{(k)}}^{(k)}\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\{a_{n_j}^{(k)}\\}\\)\ub294 \uc218\ub834\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.13.<\/span><br \/>\n\\(E=\\{x_1,x_2,x_3,\\ldots\\}\\)\uac00 \uac00\uc0b0\uc9d1\ud569\uc774\uace0 \\(\\{f_n\\}\\)\uc774 \\(E\\)\uc5d0\uc11c \\(\\mathbb R\\)\ub85c \uac00\ub294 \ud568\uc218\ub4e4\uc758 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218\uc5f4 \\(\\{f_n(x_k)\\}_{n=1}^{\\infty}\\)\uc774 \uc720\uacc4\uc774\uba74, \uc5b4\ub5a4 \ubd80\ubd84\uc218\uc5f4 \\(\\{f_{n_j}\\}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud574 \\(\\{f_{n_j}(x)\\}\\)\uac00 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\uc640 \uac19\uc740 \ub300\uac01\uc120 \uc120\ud0dd\uc740 \ub4a4\uc5d0\uc11c \ud568\uc218\uc5f4\uc758 \uc218\ub834\uc744 \ub2e4\ub8f0 \ub54c \ub2e4\uc2dc \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<\/div>\n<h3>\ucef4\ud329\ud2b8\uc131<\/h3>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\)\uac00 \uc720\uacc4\ub77c\ub294 \uac83\uc740 \uc5b4\ub5a4 \\(p\\in X\\)\uc640 \\(R&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(A\\subseteq B(p,R)\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc911\uc2ec \\(p\\)\uc758 \uc120\ud0dd\uc740 \uc911\uc694\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(K\\)\uac00 <span class=\"defined\">\ucef4\ud329\ud2b8<\/span>(compact) \uacf5\uac04\uc774\ub77c \ud568\uc740 \\(K\\)\uc758 \uc784\uc758\uc758 \uc5f4\ub9b0\ub36e\uac1c\uac00 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\ub97c \uac00\uc9c0\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc989<br \/>\n\\[<br \/>\nK\\subseteq\\bigcup_{\\alpha\\in I}U_\\alpha<br \/>\n\\]<br \/>\n\uc778 \uc5f4\ub9b0\uc9d1\ud569\ub4e4 \\(\\{U_\\alpha\\}\\)\uc5d0 \ub300\ud574, \uc720\ud55c \uac1c\uc758 \ucca8\uc790 \\(\\alpha_1,\\ldots,\\alpha_m\\)\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nK\\subseteq U_{\\alpha_1}\\cup\\cdots\\cup U_{\\alpha_m}<br \/>\n\\]<br \/>\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(K\\)\uac00 \ucef4\ud329\ud2b8\ub77c\ub294 \ub9d0\uc740 \ubd80\ubd84\uacf5\uac04\uc73c\ub85c\uc11c \ucef4\ud329\ud2b8\ud558\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.14.<\/span><br \/>\n\\(A\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc774\uba74 \\(A\\)\ub294 \ucef4\ud329\ud2b8\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.10. (\ucef4\ud329\ud2b8 \uc9d1\ud569\uc758 \uae30\ubcf8\uc131\uc9c8)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569 \\(K\\)\ub294 \uc720\uacc4\uc774\uace0 \\(X\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(K\\)\uac00 \uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc790. \\(K=\\varnothing\\)\uc774\uba74 \uc790\uba85\ud558\ubbc0\ub85c \\(K\\neq\\varnothing\\)\uc774\ub77c \ud558\uace0 \\(p\\in K\\)\ub97c \ud558\ub098 \uace0\ub974\uc790. \\(\\{B(p,n)\\}_{n\\in\\mathbb N}\\)\uc740 \\(K\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\ubbc0\ub85c \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \\(N\\)\uc5d0 \ub300\ud558\uc5ec \\(K\\subseteq B(p,N)\\)\uc774\uace0 \\(K\\)\ub294 \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\ub2eb\ud798\ub3c4 \\(K=\\varnothing\\)\uc774\uba74 \uc790\uba85\ud558\ubbc0\ub85c \uacc4\uc18d\ud574\uc11c \\(K\\neq\\varnothing\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(x\\in X\\setminus K\\)\ub77c\uace0 \ud558\uc790. \uac01 \\(y\\in K\\)\uc5d0 \ub300\ud558\uc5ec \\(r_y=d(x,y)\/2&gt;0\\)\ub85c \ub450\uba74 \\(\\{B(y,r_y)\\}_{y\\in K}\\)\ub294 \\(K\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\ub2e4. \ucef4\ud329\ud2b8\uc131\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\nK\\subseteq B(y_1,r_{y_1})\\cup\\cdots\\cup B(y_m,r_{y_m})<br \/>\n\\]<br \/>\n\uc778 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\delta=\\min\\{r_{y_1},\\ldots,r_{y_m}\\}&gt;0\\)\ub85c \ub450\uc790. \\(z\\in K\\)\uc774\uba74 \uc5b4\ub5a4 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(z\\in B(y_i,r_{y_i})\\)\uc774\uace0<br \/>\n\\[<br \/>\nd(x,z)\\geq d(x,y_i)-d(y_i,z)&gt;2r_{y_i}-r_{y_i}=r_{y_i}\\geq\\delta.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(B(x,\\delta)\\cap K=\\varnothing\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(X\\setminus K\\)\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(K\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.15.<\/span><br \/>\n\\(K\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(F\\)\uac00 \\(X\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ub2eb\ud78c \ubd80\ubd84\uc9d1\ud569\uc774\uba70 \\(K\\cap F=\\varnothing\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub450 \uc9d1\ud569 \uc0ac\uc774\uc758 \uac70\ub9ac\ub97c \uc815\uc758\ud558\uc790.<br \/>\n\\[<br \/>\nd(K,F)=\\inf\\{d(x,y)\\mid x\\in K,\\ y\\in F\\}.<br \/>\n\\]<br \/>\n\uc774\ub54c \\(d(K,F)&gt;0\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.16.<\/span><br \/>\n\ucef4\ud329\ud2b8 \uacf5\uac04\uc758 \ub2eb\ud78c \ubd80\ubd84\uc9d1\ud569\uc774 \ucef4\ud329\ud2b8\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04 \\(K\\)\uac00 <span class=\"defined\">\uc810\uc5f4\ucef4\ud329\ud2b8<\/span>(sequentially compact) \uacf5\uac04\uc774\ub77c \ud568\uc740 \\(K\\)\uc758 \ubaa8\ub4e0 \uc218\uc5f4\uc774 \\(K\\)\uc758 \uc810\uc5d0 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac00\uc9c0\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \ub610\ud55c \uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 <span class=\"defined\">\uc804\uc720\uacc4<\/span>(totally bounded)\ub77c\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc720\ud55c \uac1c\uc758 \uc810 \\(x_1,\\ldots,x_m\\in X\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nX\\subseteq\\bigcup_{j=1}^mB(x_j,\\varepsilon)<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.11. (\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\uc758 \uc810\uc5f4\ucef4\ud329\ud2b8\uc131)<\/span><\/p>\n<p>\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \ucef4\ud329\ud2b8\uc131\uacfc \uc810\uc5f4\ucef4\ud329\ud2b8\uc131\uc740 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(X=\\varnothing\\)\uc774\uba74 \ub450 \uc131\uc9c8\uc774 \ubaa8\ub450 \uc790\uba85\ud558\ubbc0\ub85c \\(X\\neq\\varnothing\\)\uc774\ub77c\uace0 \ud558\uc790. \uba3c\uc800 \\(X\\)\uac00 \ucef4\ud329\ud2b8\ub77c\uace0 \ud558\uc790. \uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(r_x&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B(x,r_x)\\)\uc5d0 \uc18d\ud558\ub294 \ud56d\uc740 \uc720\ud55c \uac1c\ubfd0\uc774\ub2e4. \uadf8\ub807\uc9c0 \uc54a\uc73c\uba74 \\(B(x,1\/k)\\)\uc5d0 \ub4e4\uc5b4\uac00\ub294 \ud56d\uc744 \ucca8\uc790\uac00 \uc99d\uac00\ud558\ub3c4\ub85d \ud558\ub098\uc529 \uace8\ub77c \\(x\\)\ub85c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \ub9cc\ub4e4 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4. \\(\\{B(x,r_x)\\}_{x\\in X}\\)\ub294 \\(X\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\ubbc0\ub85c \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uc720\ud55c \uac1c\uc758 \uacf5 \uac01\uac01\uc5d0 \ub4e4\uc5b4\uac00\ub294 \ud56d\uc758 \ucca8\uc790\uac00 \uc720\ud55c \uac1c\ubfd0\uc774\uba74 \uc790\uc5f0\uc218 \uc804\uccb4\uac00 \uc720\ud55c \uac1c\uc758 \uc720\ud55c\uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc774 \ub418\uc5b4\uc57c \ud558\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(X\\)\ub294 \uc810\uc5f4\ucef4\ud329\ud2b8\uc774\ub2e4.<\/p>\n<p>\uc5ed\uc73c\ub85c \\(X\\)\uac00 \uc810\uc5f4\ucef4\ud329\ud2b8\ub77c\uace0 \ud558\uc790. \uba3c\uc800 \\(X\\)\uac00 \uc804\uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc790. \uc804\uc720\uacc4\uac00 \uc544\ub2c8\uba74 \uc5b4\ub5a4 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc720\ud55c \uac1c\uc758 \ubc18\uc9c0\ub984 \\(\\varepsilon\\)\uc778 \uc5f4\ub9b0\uacf5\uc73c\ub85c \\(X\\)\ub97c \ub36e\uc744 \uc218 \uc5c6\ub2e4. \\(x_1\\in X\\)\ub97c \ud0dd\ud558\uace0, \uc774\ubbf8 \\(x_1,\\ldots,x_n\\)\uc744 \ud0dd\ud588\ub2e4\uba74<br \/>\n\\[<br \/>\nx_{n+1}\\notin\\bigcup_{j=1}^nB(x_j,\\varepsilon)<br \/>\n\\]<br \/>\n\uac00 \ub418\ub3c4\ub85d \ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74 \\(m\\neq n\\)\uc77c \ub54c \\(d(x_m,x_n)\\geq\\varepsilon\\)\uc774\ubbc0\ub85c \uc774 \uc218\uc5f4\uc740 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac00\uc9c8 \uc218 \uc5c6\ub2e4. \uc774\ub294 \uc810\uc5f4\ucef4\ud329\ud2b8\uc131\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\mathcal U\\)\uac00 \\(X\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\ub77c\uace0 \ud558\uc790. \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in X\\)\uc5d0 \ub300\ud574 \\(B(x,\\delta)\\)\uac00 \\(\\mathcal U\\)\uc758 \uc5b4\ub5a4 \uc6d0\uc18c\uc5d0 \ud3ec\ud568\ub428\uc744 \ubcf4\uc774\uc790. \uadf8\ub807\uc9c0 \uc54a\uc73c\uba74 \uac01 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x_n,1\/n)\\)\uc774 \\(\\mathcal U\\)\uc758 \uc5b4\ub290 \uc6d0\uc18c\uc5d0\ub3c4 \ud3ec\ud568\ub418\uc9c0 \uc54a\ub294 \uc810 \\(x_n\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \uc810\uc5f4\ucef4\ud329\ud2b8\uc131\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \ubd80\ubd84\uc218\uc5f4 \\(x_{n_j}\\to x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(x\\in U\\in\\mathcal U\\)\uc778 \\(U\\)\ub97c \ud0dd\ud558\uba74, \\(U\\)\uac00 \uc5f4\ub824 \uc788\uc73c\ubbc0\ub85c \uc5b4\ub5a4 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\subseteq U\\)\uc774\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(j\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd(x_{n_j},x)&lt;\\frac r2,\\quad \\frac1{n_j}&lt;\\frac r2<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(B(x_{n_j},1\/n_j)\\subseteq B(x,r)\\subseteq U\\)\uac00 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uc804\uc720\uacc4\uc131\uc5d0 \uc758\ud574 \uc720\ud55c \uac1c\uc758 \uc810 \\(y_1,\\ldots,y_m\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nX\\subseteq\\bigcup_{i=1}^mB\\left(y_i,\\frac\\delta2\\right)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uac01 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(B(y_i,\\delta)\\subseteq U_i\\)\uc778 \\(U_i\\in\\mathcal U\\)\ub97c \ud0dd\ud558\uba74 \\(U_1,\\ldots,U_m\\)\uc774 \\(X\\)\ub97c \ub36e\ub294\ub2e4. \ub530\ub77c\uc11c \\(X\\)\ub294 \ucef4\ud329\ud2b8\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.17.<\/span><br \/>\n\\(X\\)\uc640 \\(Y\\)\uac00 \ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">2\uc7a5<\/a>\uc758 \uacf1\uac70\ub9ac \uc911 \ud558\ub098\ub97c \uc900 \\(X\\times Y\\)\uac00 \ucef4\ud329\ud2b8\uc784\uc744 \uc815\ub9ac 3.11\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.18.<\/span><br \/>\n\ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774 \uc644\ube44\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \ucef4\ud329\ud2b8 \uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc644\uc804\ud788 \ud2b9\uc131\ud654\ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.12. (\ud558\uc774\ub124-\ubcf4\ub810)<\/span><\/p>\n<p>\\(\\mathbb R^n\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774 \ucef4\ud329\ud2b8\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub2eb\ud600 \uc788\uace0 \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \ub2eb\ud600 \uc788\uace0 \uc720\uacc4\uc778 \uac83\uc740 \uc815\ub9ac 3.10\uc5d0\uc11c \uc99d\uba85\ud558\uc600\ub2e4. \uc774\uc81c \\(K\\subseteq\\mathbb R^n\\)\uc774 \ub2eb\ud600 \uc788\uace0 \uc720\uacc4\ub77c\uace0 \ud558\uc790. \\(K\\)\uac00 \ucef4\ud329\ud2b8\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uba74 \\(K\\)\uc758 \uc5b4\ub5a4 \uc5f4\ub9b0\ub36e\uac1c \\(\\mathcal U\\)\ub294 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\\(K\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(M&gt;0\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nK\\subseteq Q_0=[-M,M]^n<br \/>\n\\]<br \/>\n\uc774\ub2e4. \\(Q_0\\)\uc758 \uac01 \uc88c\ud45c\uad6c\uac04\uc744 \uc774\ub4f1\ubd84\ud558\uba74 \\(2^n\\)\uac1c\uc758 \ub2eb\ud78c \uc815\uc721\uba74\uccb4\ub85c \ub098\ub25c\ub2e4. \uadf8\uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(K\\)\uc640\uc758 \uad50\uc9d1\ud569\uc774 \\(\\mathcal U\\)\uc758 \uc720\ud55c \uac1c \uc6d0\uc18c\ub85c \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ud55c \uc815\uc721\uba74\uccb4\ub97c \\(Q_1\\)\ub85c \ud0dd\ud55c\ub2e4. \uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74<br \/>\n\\[<br \/>\nQ_0\\supseteq Q_1\\supseteq Q_2\\supseteq\\cdots<br \/>\n\\]<br \/>\n\uc778 \ub2eb\ud78c \uc815\uc721\uba74\uccb4\uc758 \uc5f4\uc744 \uc5bb\uc73c\uba70, \uac01 \\(Q_k\\cap K\\)\ub294 \\(\\mathcal U\\)\uc758 \uc720\ud55c \uac1c \uc6d0\uc18c\ub85c \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \ud2b9\ud788 \\(Q_k\\cap K\\neq\\varnothing\\)\uc774\uace0, \\(Q_k\\)\uc758 \ud55c \ubcc0\uc758 \uae38\uc774\ub294 \\(2M\/2^k\\)\uc774\ub2e4.<\/p>\n<p>\uac01 \uc88c\ud45c\uc5d0\uc11c \uc0dd\uae30\ub294 \ub2eb\ud78c\uad6c\uac04\ub4e4\uc740 \ucd95\uc18c\uad6c\uac04\uc758 \uc5f4\uc744 \uc774\ub8e8\ubbc0\ub85c <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc815\ub9ac 1.5\uc758 \ucd95\uc18c\uad6c\uac04 \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \\(Q_k\\)\uc5d0 \uc18d\ud558\ub294 \uc810 \\(x\\in\\mathbb R^n\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \uac01 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(x_k\\in Q_k\\cap K\\)\ub97c \ud0dd\ud558\uba74<br \/>\n\\[<br \/>\nd_2(x_k,x)\\leq\\frac{2M\\sqrt n}{2^k}\\longrightarrow0.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(x_k\\to x\\)\uc774\uace0, \\(K\\)\uac00 \ub2eb\ud600 \uc788\uc73c\ubbc0\ub85c \\(x\\in K\\)\uc774\ub2e4.<\/p>\n<p>\\(\\mathcal U\\)\uac00 \\(K\\)\ub97c \ub36e\uc73c\ubbc0\ub85c \\(x\\in U\\)\uc778 \\(U\\in\\mathcal U\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(U\\)\uac00 \uc5f4\ub824 \uc788\uc73c\ubbc0\ub85c \uc5b4\ub5a4 \\(r&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(x,r)\\subseteq U\\)\uc774\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0\uc11c\ub294 \\(2M\\sqrt n\/2^k&lt;r\\)\uc774\uace0 \\(x\\in Q_k\\)\uc774\ubbc0\ub85c \\(Q_k\\subseteq B(x,r)\\subseteq U\\)\uc774\ub2e4. \uadf8\ub7ec\uba74 \\(Q_k\\cap K\\)\uac00 \\(U\\) \ud558\ub098\ub85c \ub36e\uc5ec \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(K\\)\ub294 \ucef4\ud329\ud2b8\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.19.<\/span><br \/>\n\\([0,1]\\)\uc5d0\uc11c \uac00\uc6b4\ub370 \uc5f4\ub9b0 \uc0bc\ubd84\uc758 \uc77c\uc744 \ubc18\ubcf5\ud558\uc5ec \uc81c\uac70\ud558\uc5ec \uc5bb\ub294 \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc744 \\(C\\)\ub77c\uace0 \ud558\uc790. \uc989 \\(C_0=[0,1]\\)\uc774\uace0, \\(C_{m+1}\\)\uc740 \\(C_m\\)\uc758 \uac01 \ub2eb\ud78c\uad6c\uac04\uc5d0\uc11c \uac00\uc6b4\ub370 \uc5f4\ub9b0 \uc0bc\ubd84\uc758 \uc77c\uc744 \uc81c\uac70\ud558\uc5ec \uc5bb\uc73c\uba70<br \/>\n\\[<br \/>\nC=\\bigcap_{m=0}^{\\infty}C_m<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(C\\)\uac00 \ub2eb\ud600 \uc788\uace0 \uc720\uacc4\uc774\ubbc0\ub85c \ucef4\ud329\ud2b8\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\uc11c\ub85c \ub2e4\ub978 \\(x,y\\in C\\) \uc0ac\uc774\uc5d0\ub294 \\(C\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uc810\uc774 \ud56d\uc0c1 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc2e4\uc9c1\uc120\uc758 \ubd80\ubd84\uc9d1\ud569\uc5d0 \ub300\ud574\uc11c\ub294 \uc774 \uc131\uc9c8\uc774 \uc644\uc804\ubd88\uc5f0\uacb0\uc131(total disconnectedness)\uc744 \ub73b\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.20.<\/span><br \/>\n\uc9d1\ud569 \\(E\\)\uac00 \\(\\mathbb R^n\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\{U_\\alpha\\}_{\\alpha\\in A}\\)\uac00 \\(E\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\uba74 \uac00\uc0b0 \uac1c\uc758 \ucca8\uc790 \\(\\alpha_1,\\alpha_2,\\ldots\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nE\\subseteq\\bigcup_{k=1}^{\\infty}U_{\\alpha_k}<br \/>\n\\]<br \/>\n\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \uc774\uac83\uc740 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc774 <span class=\"defined\">\ub9b0\ub378\ub8b0\ud504 \uacf5\uac04<\/span>(Lindel\u00f6f space)\uc784\uc744 \ub73b\ud55c\ub2e4. \\(\\mathbb Q^n\\)\uc774 \uac00\uc0b0\uc774\uace0 \uc870\ubc00\ud558\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.21.<\/span><br \/>\n\\(G\\)\uac00 \\(\\mathbb R\\)\uc758 \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(G\\)\ub294 \uc30d\ub9c8\ub2e4 \uc11c\ub85c\uc18c\uc778 \uac00\uc0b0 \uac1c \uc774\ud558\uc758 \uc5f4\ub9b0\uad6c\uac04\ub4e4\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624. \uc5ec\uae30\uc11c \uc591 \ub05d\uc810\uc774 \ubb34\ud55c\ub300\uc778 \uc5f4\ub9b0\uad6c\uac04\ub3c4 \ud5c8\uc6a9\ud55c\ub2e4. (\uc774 \uc131\uc9c8\uc740 \uc2e4\uc9c1\uc120\uc5d0\uc11c \ub974\ubca0\uadf8 \uce21\ub3c4\ub97c \uc815\uc758\ud560 \ub54c \uc0ac\uc6a9\ub41c\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.22.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(p\\in X\\), \\(r&gt;0\\)\uc77c \ub54c \\(\\overline B(p,r)=\\overline{B(p,r)}\\)\uc774 \uc131\ub9bd\ud558\ub294\uac00? \uc131\ub9bd\ud55c\ub2e4\uba74 \uc99d\uba85\ud558\uace0, \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.23.<\/span><br \/>\n\ud45c\uc900\uac70\ub9ac \\(d(m,n)=|m-n|\\)\uac00 \uc8fc\uc5b4\uc9c4 \uc815\uc218 \uc9d1\ud569 \\(\\mathbb Z\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \uac70\ub9ac\uacf5\uac04\uc758 \uc644\ube44\uc131\uc744 \ub2e8\uc21c\ud788 \u201c\uc9c1\uc120\uc744 \ube48\ud2c8 \uc5c6\uc774 \uac00\ub4dd \ucc44\uc6b4 \uac83\u201d\uc774\ub77c\uace0 \uc124\uba85\ud558\ub294 \ube44\uc720\uac00 \uc65c \uc815\ud655\ud558\uc9c0 \uc54a\uc740\uc9c0 \uc124\uba85\ud558\uace0, \ucf54\uc2dc \uc218\uc5f4\uc744 \uc774\uc6a9\ud55c \ub300\uc548\uc744 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.24.<\/span><br \/>\n\uc9d1\ud569 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784 \\(\\mathcal T\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\((X,\\mathcal T)\\)\ub97c <span class=\"defined\">\uc704\uc0c1\uacf5\uac04<\/span>(topological space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(X,\\varnothing\\in\\mathcal T\\).<\/li>\n<li>\\(\\mathcal T\\)\uc758 \uc6d0\uc18c\ub4e4\uc758 \uc784\uc758\uc758 \ud569\uc9d1\ud569\ub3c4 \\(\\mathcal T\\)\uc5d0 \uc18d\ud55c\ub2e4.<\/li>\n<li>\\(\\mathcal T\\)\uc758 \uc6d0\uc18c\ub4e4\uc758 \uc720\ud55c \uad50\uc9d1\ud569\ub3c4 \\(\\mathcal T\\)\uc5d0 \uc18d\ud55c\ub2e4.<\/li>\n<\/ol>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubaa8\ub4e0 \uc5f4\ub9b0\uc9d1\ud569\uc758 \ubaa8\uc784\uc774 \uc704\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \uc815\ub9ac 3.3\uc744 \uc774\uc6a9\ud558\uc5ec \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.25.<\/span><br \/>\n\uc9d1\ud569 \\(X\\)\uc5d0 \uac70\ub9ac\ud568\uc218 \\(d_1\\), \\(d_2\\)\uac00 \uc8fc\uc5b4\uc838 \uc788\uace0, \uac70\ub9ac\uacf5\uac04 \\((X,d_1)\\)\uacfc \\((X,d_2)\\)\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc758 \ubaa8\uc784\uc744 \uac01\uac01 \\(T_1,T_2\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(T_1=T_2\\)\uc774\uba74 \u201c\ub450 \uac70\ub9ac\ud568\uc218 \\(d_1\\)\uacfc \\(d_2\\)\uac00 \ub3d9\uce58\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc591\uc218 \\(k_1,k_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nk_1d_1(x,y)\\leq d_2(x,y)\\leq k_2d_1(x,y)<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(d_1\\)\uacfc \\(d_2\\)\uac00 \ub3d9\uce58\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(\\lVert\\cdot\\rVert_1\\)\uacfc \\(\\lVert\\cdot\\rVert_2\\)\uac00 \\(\\mathbb R^n\\)\uc758 \ub178\ub984\uc77c \ub54c, \ub450 \ub178\ub984\uc73c\ub85c\ubd80\ud130 \uc720\ub3c4\ub41c \uac70\ub9ac \\(d_1\\)\uacfc \\(d_2\\)\uac00 \ub3d9\uce58\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\uc720\ud55c \uac1c\uc758 \ud56d\ub9cc 0\uc774 \uc544\ub2cc \uc2e4\uc218\uc5f4\ub4e4\uc758 \ubca1\ud130\uacf5\uac04\uc744 \\(c_{00}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(c_{00}\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\|x\\|_1=\\sum_{k=1}^{\\infty}|x_k|,\\quad<br \/>\n\\|x\\|_\\infty=\\sup_{k\\geq1}|x_k|<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uba74 \ub450 \ub178\ub984\uc774 \uc720\ub3c4\ud558\ub294 \uac70\ub9ac\uac00 \ub3d9\uce58\uac00 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \ud1b5\ud558\uc5ec \ubb34\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc5d0\uc11c\ub294 \uc11c\ub85c \ub3d9\uce58\uac00 \uc544\ub2cc \ub178\ub984\uc774 \uc874\uc7ac\ud560 \uc218 \uc788\uc74c\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n<ol class=\"contentboxorderedlist\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\uac70\ub9ac\uacf5\uac04<\/a><\/li>\n<li class=\"contentboxthis\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\">\uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc911\uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\">\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>2\uc7a5\uc5d0\uc11c \uac70\ub9ac\uacf5\uac04\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uacfc \ucf54\uc2dc \uc870\uac74\uc744 \uc77c\ubc18 \uac70\ub9ac\uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ud3d0\ud3ec, \ucef4\ud329\ud2b8\uc131\ucc98\ub7fc \uac70\ub9ac\ub85c\ubd80\ud130 \uc0dd\uae30\ub294 \uc704\uc0c1\uc801 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\uc758 \uc218\ub834 \uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc5d0\uc11c \uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \uc810 \\(x\\in X\\)\ub85c \uc218\ub834\ud55c\ub2e4(converge)\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n&gt;N\\)\uc77c \ub54c \\(d(x_n,x)&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc774\uac83\uc744 \\(x_n\\to x\\) \ub610\ub294 \\(\\lim_{n\\to\\infty}x_n=x\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc815\ub9ac 3.1. (\uadf9\ud55c\uc758 \uc720\uc77c\uc131) \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uc720\uc77c\ud558\ub2e4. \uc99d\uba85 \uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \\(x\\)\uc640 \\(y\\)\uc5d0 \ubaa8\ub450&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":103,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9480","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9480","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9480"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9480\/revisions"}],"predecessor-version":[{"id":10105,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9480\/revisions\/10105"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}