{"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":"2025-10-21T16:07:50","modified_gmt":"2025-10-21T07:07:50","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"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\uc218\uc5f4\uc758 \uadf9\ud55c<\/h2>\n\n --><\/p>\n<p>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834\uacfc \uae30\ubcf8\uc801\uc778 \uc704\uc0c1 \uac1c\ub150\uc744 \ub2e4\ub8ec\ub2e4. \uc2e4\uc218\uc5d0\uc11c \uc815\uc758\ud55c \uadf9\ud55c \uac1c\ub150\uc744 \uc77c\ubc18 \uac70\ub9ac\uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ucef4\ud329\ud2b8\uc131 \ub4f1 \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 > 0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\(d(x_n,\\, x) < \\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<p>\uc218\ub834\ud558\ub294 \uc2e4\uc218\uc5f4\uc758 \uadf9\ud55c\uc774 \uc720\uc77c\ud55c \uac83\ucc98\ub7fc, \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\ub3c4 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uc720\uc77c\ud558\ub2e4. \uc989, \\(x_n \\to x\\)\uc774\uace0 \\(x_n \\to y\\)\uc774\uba74 \\(x = y\\)\uc774\ub2e4.<\/p>\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 > 0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m >N\\), \\(n > N\\)\uc77c \ub54c \\(d(x_m,\\, x_n) < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<p>\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \ud56d\uc0c1 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc, \uadf8 \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \ubaa8\ub4e0 \ucf54\uc2dc \uc218\uc5f4\uc774 \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.<\/p>\n<ul>\n<li>\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uac00 \uc8fc\uc5b4\uc9c4 \uacf5\uac04 \\(\\mathbb{R}^n\\)\uc740 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\ub2e4.<\/li>\n<li>\ud45c\uc900 \uac70\ub9ac [\\(d(x,\\,y)=|x-y|\\)\ub85c \uc815\uc758\ub41c \uac70\ub9ac\ud568\uc218] \uac00 \uc8fc\uc5b4\uc9c4 \uc720\ub9ac\uc218 \\(\\mathbb{Q}\\)\ub294 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(\\sqrt{2}\\)\ub85c \uc218\ub834\ud558\ub294 \uc720\ub9ac\uc218\uc5f4\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc \\(\\mathbb{Q}\\)\uc5d0\uc11c \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. \uc65c\ub0d0\ud558\uba74, \uc218\uc5f4 \\(\\left\\{\\frac{1}{n}\\right\\}\\)\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc \\((0,\\, 1)\\)\uc758 \uc810\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.1.<\/span><br \/>\n\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(\\mathbb{R}^d\\)\uc758 \uc810\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774\uace0, \\(\\left\\{ a_n \\right\\}\\)\uc758 \uac01 \ud56d\uc774<br \/>\n\\[a_n = \\left( a_{n,1},\\, a_{n,2},\\, \\cdots ,\\, a_{n,d}\\right)\\]<br \/>\n\uc640 \uac19\uc740 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(L = (L_1 ,\\, L_2 ,\\, \\cdots ,\\, L_d)\\in \\mathbb{R}^d\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c, \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(j=1,\\,2,\\,\\cdots,\\,d\\)\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218\uc5f4 \\(\\left\\{ a_{n,j}\\right\\}\\)\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>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(x\\)\uac00 \\(E\\)\uc758 \uacbd\uacc4\uc810\uc774\ub77c \ud568\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\uc744 \ub73b\ud55c\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\\)\uc640 \uac19\uc774 \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)\\)\uc640 \uac19\uc774 \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\\)\uc640 \uac19\uc774 \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 3.1. (\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=\"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\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A=A^o\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(B\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc77c \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 &#8216;\uc5f4\ub9b0\uadfc\ubc29&#8217;\uc744 &#8216;\uadfc\ubc29&#8217;\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, &#8220;\uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(p(x)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4&#8221;\ub77c\ub294 \ub9d0\uc740 &#8220;\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&#8221;\ub77c\ub294 \ub73b\uc774\ub2e4. [&#8220;\uc810 \\(a\\)\uc758 &#8216;\uc784\uc758\uc758&#8217; \uadfc\ubc29 \\(U\\)\uc5d0\uc11c \\(p(x)\\)\uac00 \ucc38\uc774\ub2e4&#8221;\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 \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(X\\)\uc758 \uc218\uc5f4\uc774\uba70 \\(L\\in X\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud560 \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 3.1\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 3.2. (\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=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.4.<\/span><br \/>\n\uc815\ub9ac 3.1\uacfc \uc815\ub9ac 3.2\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ud55c\ud3b8 \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 3.3. (\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\uc77c \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\uc77c \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=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.5.<\/span><br \/>\n\uc815\ub9ac 3.3\uc744 \uc99d\uba85\ud558\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>\ud3d0\ud3ec\ub294 \uc9d1\uc801\uc810\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \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 > 0\\)\uc5d0 \ub300\ud574 \\(B &#8216; (x,\\, r) \\cap A \\neq \\varnothing\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc989 \\(x\\)\uac00 \\(A\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\ub294 \uac83\uc740 \\(A\\)\uc5d0 \uc18d\ud558\uba74\uc11c \\(x\\)\uc640 \uac19\uc9c0 \uc54a\uc740 \uc810\uc744 \ubaa8\uc544\uc11c \\(x\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\n<p>\\(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 &#8216; \\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\overline{A} = A \\cup A &#8216; \\tag{3.1}\\]<\/p>\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 3.4. (\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=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.6.<\/span><br \/>\n\uc815\ub9ac 3.4\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.7.<\/span><br \/>\n\ub4f1\uc2dd (3.1)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 3.8.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(E\\)\uac00 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \ub3d9\uce58\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(E\\)\uac00 \\(X\\)\uc5d0\uc11c \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(E\\)\uc5d0 \uc18d\ud558\uace0 \\(\\left\\{ a_n \\right\\}\\)\uc774 \uc218\ub834\ud558\uba74, \\(\\left\\{ a_n \\right\\}\\)\uc758 \uadf9\ud55c\uc774 \\(E\\)\uc5d0 \uc18d\ud55c\ub2e4.<\/li>\n<\/ol>\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\uc77c \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. \uc65c\ub0d0\ud558\uba74 \uc720\ub9ac\uc218 \uc88c\ud45c\ub97c \uac00\uc9c4 \uc810\ub4e4\uc774 \uc870\ubc00\ud558\uac8c \ubd84\ud3ec\ud558\uae30 \ub54c\ubb38\uc774\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.<\/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\">\uc9d1\uc801\uc810<\/span>(cluster point) \ub610\ub294 <span class=\"defined\">\ubd80\ubd84\uc218\uc5f4\uadf9\ud55c<\/span>(subsequential limit)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.10.<\/span><br \/>\n\uc218\uc5f4 \\(\\left\\{x_n\\right\\}\\)\uc774 \\(L\\)\ub85c \uc218\ub834\ud558\uace0 \\(\\{x_{n_k}\\}\\)\uac00 \\(\\left\\{x_n\\right\\}\\)\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>\uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\)\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\varlimsup_{n \\to \\infty} a_n = \\lim_{n \\to \\infty} \\left(\\sup\\{a_k \\mid k \\geq n\\}\\right),\\quad<br \/>\n\\varliminf_{n \\to \\infty} a_n = \\lim_{n \\to \\infty} \\left(\\inf\\{a_k \\mid k \\geq n\\}\\right).\\tag{3.2}\\]<br \/>\n\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\displaystyle \\varliminf_{n\\rightarrow\\infty} a_n \\leq \\varlimsup_{n\\rightarrow\\infty} a_n .\\tag{3.3}\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.11.<\/span><br \/>\n\uc591\uc758 \uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\), \\(\\{b_n\\}\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n\\[\\varlimsup_{n \\to \\infty} (a_n b_n) \\leq \\varlimsup_{n \\to \\infty} a_n \\cdot \\varlimsup_{n \\to \\infty} b_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><span class=\"definition\">\uc815\ub9ac 3.5. (\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc758 \uae30\ubcf8\uc131\uc9c8)<\/span><\/p>\n<p>\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc720\uacc4\uc778 \uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc740 \ud56d\uc0c1 \uc2e4\uc218\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc720\uacc4\uc778 \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \uc0c1\uadf9\ud55c\uc740 \uc9d1\uc801\uc810 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc774\uace0, \\(\\left\\{ a_n \\right\\}\\)\uc758 \ud558\uadf9\ud55c\uc740 \uc9d1\uc801\uc810 \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac12\uc774\ub2e4.<\/li>\n<li>\uc720\uacc4\uc778 \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\left\\{ a_n \\right\\}\\)\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc774 \uc77c\uce58\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\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 + \\frac{1}{n}\\right)\\)\uc73c\ub85c \uc815\uc758\ub41c \uc218\uc5f4 \\(\\left\\{a_n \\right\\}\\)\uc744 \uc0dd\uac01\ud558\uc790. \uc774 \uc218\uc5f4\uc758 \uc9d1\uc801\uc810\uc740 \\(-1\\)\uacfc \\(1\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc740 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\varlimsup_{n\\rightarrow\\infty} a_n = 1 ,\\quad \\varliminf_{n\\rightarrow\\infty} a_n= -1 .\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.12.<\/span><br \/>\n\uc815\ub9ac 3.5\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc218\uc5f4\uc758 \uac01 \ud56d\uc774 \uc218\uc5f4\uc77c \uc218 \uc788\ub2e4. \uc774\ub54c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \ucc3e\ub294 \ubc29\ubc95 \uc911 \ud558\ub098\uac00 <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 3.6. (\ub300\uac01\uc120 \ub17c\ubc95)<\/span><\/p>\n<p>\uc720\uacc4\uc778 \uc218\uc5f4\uc758 \uc218\uc5f4 \\(\\{a_n^{(k)}\\}_{n=1}^{\\infty}\\) (\\(k = 1,\\, 2,\\, 3,\\, \\cdots\\))\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, &#8220;\ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\{a_n^{(k)}\\}_{n=1}^{\\infty}\\)\uc774 \uc218\ub834&#8221;\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \ucc3e\uc744 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uba3c\uc800 \ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(\\{a_n^{(1)}\\}\\)\uc758 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uc120\ud0dd\ud55c\ub2e4.<br \/>\n\uc774 \ubd80\ubd84\uc218\uc5f4\uc758 \ucca8\uc790\ub97c \uc720\uc9c0\ud558\uace0, \ub2e4\uc2dc \\(\\{a_n^{(2)}\\}\\)\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ubd80\ubd84\uc218\uc5f4\uc744 \uc120\ud0dd\ud55c\ub2e4.<br \/>\n\uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \uac01 \ub2e8\uacc4\uc5d0\uc11c \ubd80\ubd84\uc218\uc5f4\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub300\uac01\uc120 \ubd80\ubd84\uc218\uc5f4 \\(\\{a_{n_n}^{(k)}\\}\\)\uc744 \uad6c\uc131\ud558\uba74, \uc774 \ubd80\ubd84\uc218\uc5f4\uc740 \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud574 \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\uc544\uccbc\ub77c-\uc560\uc2a4\ucf5c\ub9ac \uc815\ub9ac(Arzel\u00e0\u2013Ascoli theorem)\ub97c \uc870\uc0ac\ud558\uace0, \uc544\uccbc\ub77c-\uc560\uc2a4\ucf5c\ub9ac \uc815\ub9ac\uc758 \uc99d\uba85 \uacfc\uc815\uacfc \uc815\ub9ac 3.6\uc758 \uc5f0\uad00\uc131\uc744 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\ucef4\ud329\ud2b8\uc131<\/h3>\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\\[K \\subseteq \\bigcup_{\\alpha \\in I} U_\\alpha\\]<br \/>\n\uc778 \uc5f4\ub9b0\uc9d1\ud569\ub4e4 \\(\\{U_\\alpha\\}\\)\uc5d0 \ub300\ud574, \uc720\ud55c \uac1c\uc758 \ucca8\uc790 \\(\\alpha_1,\\, \\ldots,\\, \\alpha_n\\)\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[K \\subseteq U_{\\alpha_1} \\cup \\cdots \\cup U_{\\alpha_n}\\]<br \/>\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\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=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.15.<\/span><br \/>\n\\(F\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\uba74 \\(F\\)\ub294 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c\uc9d1\ud569\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.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.7. (\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 \uac1c\uc694<\/span><br \/>\n\\(X\\)\uac00 \uc810\uc5f4\ucef4\ud329\ud2b8 \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uacf5\uac04\uc774 \uc804\uc720\uacc4(totally bounded)\uc784\uc744 \ubcf4\uc778\ub2e4. \uadf8 \ub2e4\uc74c, \uc804\uc720\uacc4\uc774\uba74\uc11c \uc644\ube44\uc778 \uacf5\uac04\uc774 \ucef4\ud329\ud2b8 \uacf5\uac04\uc784\uc744 \ubcf4\uc778\ub2e4.<\/p>\n<p>\uc5ed\uc73c\ub85c \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uacf5\uac04\uc5d0\uc11c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac16\uc9c0 \uc54a\ub294 \uc218\uc5f4\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0 \ubaa8\uc21c\uc744 \uc720\ub3c4\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.17.<\/span><br \/>\n<span class=\"defined\">\uc804\uc720\uacc4 \uacf5\uac04<\/span>(totally bounded space)\uc758 \uc815\uc758\ub97c \uc870\uc0ac\ud558\uace0, \uc815\ub9ac 3.7\uc758 \uc99d\uba85\uc744 \uc644\uc131\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\uc740 \uc720\uc6a9\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 3.8. (\ud558\uc774\ub124-\ubcf4\ub810)<\/span><\/p>\n<p>\\(\\mathbb{R}^n\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774 \ucef4\ud329\ud2b8\uc77c \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=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(n=2\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774 \ub2eb\ud600\uc788\uace0 \uc720\uacc4\uc784\uc740 \uc77c\ubc18\uc801\uc778 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc131\ub9bd\ud558\ubbc0\ub85c, \uc5ec\uae30\uc11c\ub294 \uadf8 \uc5ed\ub9cc \uc99d\uba85\ud558\uc790. \\(K\\)\uac00 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0, \ub2eb\ud600\uc788\uc73c\uba70 \uc720\uacc4\uc778 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uacb0\ub860\uacfc\ub294 \ubc18\ub300\ub85c \\(K\\)\uac00 \ucef4\ud329\ud2b8\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(K\\)\ub97c \ub36e\ub294 \uc5f4\ub9b0\uc9d1\ud569\ub4e4\uc758 \ubaa8\uc784 \\(C=\\left\\{ U_\\alpha \\mid \\alpha\\in I \\right\\}\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uc720\ud55c \uac1c\uc758 \\(U_\\alpha\\)\ub85c\ub294 \\(K\\)\ub97c \ub36e\uc744 \uc218 \uc5c6\ub2e4.<\/p>\n<p>\\(K\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \\(K\\subseteq [-M ,\\, M]^2\\)\uc778 \uc591\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc9d1\ud569 \\(S_0 = [-M ,\\, M]^2\\)\uc744 \ub124 \uac1c\uc758 \ub2eb\ud78c \uc815\uc0ac\uac01\ud615 \uc9d1\ud569<br \/>\n\\[ [-M,\\, 0]\\times [-M,\\,0] ,\\quad [-M,\\, 0]\\times [0,\\,M] ,\\quad [0,\\, M]\\times [-M,\\,0] ,\\quad [0,\\, M]\\times [0,\\,M]\\]<br \/>\n\uc73c\ub85c \ucabc\uac30\uc744 \ub54c(\uacbd\uacc4\uc758 \uc77c\ubd80\ubd84\uc740 \uacb9\uce68), \ub124 \uc815\uc0ac\uac01\ud615 \uc9d1\ud569 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(K\\)\uc640 \uad50\uc9d1\ud569\ud558\uc600\uc744 \ub54c \uc720\ud55c \uac1c\uc758 \\(U_\\alpha\\)\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ud55c \uc815\uc0ac\uac01\ud615 \uc9d1\ud569\uc744 \ud0dd\ud558\uc5ec \\(S_1\\)\uc774\ub77c\uace0 \ud558\uc790. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(S_1\\)\uc744 \ub124 \uac1c\uc758 \ub2eb\ud78c \uc815\uc0ac\uac01\ud615 \uc9d1\ud569\uc73c\ub85c \ucabc\uac30\uc744 \ub54c, \ub124 \uc815\uc0ac\uac01\ud615 \uc9d1\ud569 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(K\\)\uc640 \uad50\uc9d1\ud569\ud558\uc600\uc744 \ub54c \uc720\ud55c \uac1c\uc758 \\(U_\\alpha\\)\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ud55c \uc815\uc0ac\uac01\ud615 \uc9d1\ud569\uc744 \ud0dd\ud558\uc5ec \\(S_2\\)\ub77c\uace0 \ud558\uc790. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \ub2eb\ud78c \uc815\uc0ac\uac01\ud615\uc758 \uc5f4 \\(\\left\\{ E_k \\right\\}\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\\(E_k\\)\ub294 \ub2eb\ud78c \uc815\uc0ac\uac01\ud615 \uc9d1\ud569\uc774\ubbc0\ub85c \\(E_k = [a_k ,\\, b_k]\\times [c_k ,\\,d_k]\\) \uaf34\ub85c \uc4f8 \uc218 \uc788\ub2e4. \uadf8\ub7f0\ub370 \\(k\\rightarrow\\infty\\)\uc77c \ub54c \\(E_k\\)\uc758 \uac00\ub85c \uae38\uc774\uc640 \uc138\ub85c \uae38\uc774\uac00 \ubaa8\ub450 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0, \\(E_k \\supseteq E_{k+1}\\)\uc774\ubbc0\ub85c, \ubaa8\ub4e0 \\(E_k\\)\uc5d0 \uc18d\ud558\ub294 \uc810\uc740 \ub2e8 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc810\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790. \uac01 \\(E_k\\)\uc5d0\uc11c \\(K\\)\uc758 \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ \\lambda_k \\right\\}\\)\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \\(K\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \\(\\lambda\\in K\\)\uc774\ub2e4. \\(\\left\\{ U_\\alpha \\right\\}\\)\uac00 \\(K\\)\ub97c \ub36e\uc73c\ubbc0\ub85c \\(\\lambda \\in U_\\beta\\)\uc778 \\(\\beta\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(U_\\beta\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \uc815\uc0ac\uac01\ud615 \uc9d1\ud569 \\(E_k\\)\uc758 \ubcc0\uc758 \uae38\uc774\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(E_k \\subseteq U_\\beta\\)\uc778 \\(E_k\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\uc740 \\(E_k \\cap K\\)\uac00 \uc720\ud55c \uac1c\uc758 \\(U_\\alpha\\)\uc5d0 \uc758\ud558\uc5ec \ub36e\uc774\uc9c0 \uc54a\ub294\ub2e4\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\\(n=1\\) \ub610\ub294 \\(n > 2\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.19.<\/span><br \/>\n\uce78\ud1a0\uc5b4 \uc9d1\ud569(Cantor ternary set)\uc758 \uc131\uc9c8\uc744 \uc870\uc0ac\ud558\uc2dc\uc624. \uc774 \uc9d1\ud569\uc774 \ucef4\ud329\ud2b8\uc774\uba74\uc11c <span class=\"defined\">\uc644\uc804\ubd88\uc5f0\uacb0<\/span>(totally disconnected)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\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 \\(\\left\\{ U_\\alpha \\right\\}_{\\alpha\\in A}\\)\uac00 \\(E\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\uba74 \uac00\uc0b0 \uac1c\uc758 \ucca8\uc790 \\(\\alpha_1\\), \\(\\alpha_2\\), \\(\\cdots\\), \\(\\alpha_k\\), \\(\\cdots\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\(E\\subseteq U_{\\alpha_1}\\cup U_{\\alpha_2}\\cup U_{\\alpha_3} \\cup \\cdots \\cup U_{\\alpha_k} \\cup \\cdots\\)\ub97c \ub9cc\uc871\uc2dc\ud0b4\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.<\/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\uc758 \uc5f4\ub9b0\uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624. (\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>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\uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131\uc744 &#8220;\uc9c1\uc120\uc744 \ube48 \ud2c8 \uc5c6\uc774 \uac00\ub4dd \ucc44\uc6b4 \uac83&#8221;\uc774\ub77c\uace0 \ube44\uc720\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc815\uc218\uacc4\uac00 \uc644\ube44\uc784\uc744 \ubcf4\uc774\uace0, \uc774\uc640 \uac19\uc740 \ube44\uc720\uc758 \ub300\uc548\uc744 \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 3.24.<\/span><br \/>\n\uc704\uc0c1\uacf5\uac04(topological space)\uc758 \uc815\uc758\ub97c \uc870\uc0ac\ud558\uace0, \uc704\uc0c1\uacf5\uac04\uc758 \uc815\uc758\uc640 \uc815\ub9ac 3.1\uc758 \uad00\uacc4\ub97c \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 &#8220;\ub450 \uac70\ub9ac\ud568\uc218 \\(d_1\\)\uacfc \\(d_2\\)\uac00 \ub3d9\uce58\uc774\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(d_1\\)\uacfc \\(d_2\\)\uac00 \ub3d9\uce58\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc591\uc218 \\(k_1\\), \\(k_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,\\,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(k_1 d_1 (x,\\,y)\\le d_2 (x,\\,y) \\le k_2 d_1 (x,\\,y)\\)\ub97c \ub9cc\uc871\ud558\ub294 \uac83\uc784\uc744 \uc99d\uba85\ud558\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>\\(V\\)\uac00 \ubb34\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774\uace0 \\(\\lVert \\cdot \\rVert_1\\)\uacfc \\(\\lVert \\cdot \\rVert_2\\)\uac00 \\(V\\)\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\uc778\uac00? \ub3d9\uce58\ub77c\uba74 \uc99d\uba85\ud558\uace0, \uadf8\ub807\uc9c0 \uc54a\ub2e4\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\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 \ud575\uc2ec\uc815\ub9ac \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<\/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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834\uacfc \uae30\ubcf8\uc801\uc778 \uc704\uc0c1 \uac1c\ub150\uc744 \ub2e4\ub8ec\ub2e4. \uc2e4\uc218\uc5d0\uc11c \uc815\uc758\ud55c \uadf9\ud55c \uac1c\ub150\uc744 \uc77c\ubc18 \uac70\ub9ac\uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ucef4\ud329\ud2b8\uc131 \ub4f1 \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 > 0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\(d(x_n,\\, x) < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc774\uac83\uc744 \\(x_n \\to x\\) \ub610\ub294 \\(\\lim_{n \\to \\infty} x_n =&hellip;\n<\/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":8,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9480\/revisions"}],"predecessor-version":[{"id":9608,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9480\/revisions\/9608"}],"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}]}}