{"id":9473,"date":"2025-10-20T17:16:11","date_gmt":"2025-10-20T08:16:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9473"},"modified":"2026-09-27T14:33:27","modified_gmt":"2026-09-27T05:33:27","slug":"ch01-real-number-system","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\/","title":{"rendered":"\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/h2>\n\n --><\/p>\n<p>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\uc218\uacc4\ub97c \uc644\ube44\uc21c\uc11c\uccb4\ub85c \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>\uc21c\uc11c\uccb4<\/h3>\n<p>\uc9d1\ud569 \\(F\\)\uc5d0 \ub367\uc148\uacfc \uacf1\uc148\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uc774 \uc5f0\uc0b0\uc774 \ub2e4\uc74c \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(F\\)\ub97c <span class=\"defined\">\uccb4<\/span>(field)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>(F1) \ub367\uc148: \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0 0\uc774 \uc874\uc7ac\ud558\uba70, \uc784\uc758\uc758 \uc6d0\uc18c\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>(F2) \uacf1\uc148: \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0 1\uc774 \uc874\uc7ac\ud558\uba70, 0\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc6d0\uc18c\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc874\uc7ac\ud55c\ub2e4. \ud2b9\ud788 \\(1\\ne 0\\)\uc774\ub2e4.<\/li>\n<li>(F3) \ubd84\ubc30\ubc95\uce59: \uc784\uc758\uc758 \uc6d0\uc18c \\(a\\), \\(b\\), \\(c\\)\uc5d0 \ub300\ud558\uc5ec \\(a(b+c)=ab+ac\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<\/ol>\n<p>\\(b\\)\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \\(-b\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(a+(-b)\\)\ub97c \uac04\ub2e8\ud788 \\(a-b\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(b\\neq0\\)\uc77c \ub54c, \\(b\\)\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \\(\\frac{1}{b}\\) \ub610\ub294 \\(1\/b\\) \ub610\ub294 \\(b^{-1}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(a\\)\uc640 \\(\\frac{1}{b}\\)\uc758 \uacf1\uc744 \\(\\frac{a}{b}\\) \ub610\ub294 \\(a\/b\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.1.<\/span><br \/>\n\\(F\\)\uac00 \uccb4\uc77c \ub54c \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F\\)\uc5d0\uc11c \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uacfc \uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uc774 \uac01\uac01 \uc720\uc77c\ud558\ub2e4.<\/li>\n<li>\\(a\\in F\\)\uc77c \ub54c \\(a\\)\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc720\uc77c\ud558\ub2e4.<\/li>\n<li>\\(b\\in F\\)\uc774\uace0 \\(b\\ne0\\)\uc77c \ub54c \\(b\\)\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc720\uc77c\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uccb4 \\(F\\)\uc5d0 \uc21c\uc11c \uad00\uacc4 \\(\\leq\\)\uac00 \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \ub2e4\uc74c \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(F\\)\ub97c <span class=\"defined\">\uc21c\uc11c\uccb4<\/span>(ordered field)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>(O1) \uc784\uc758\uc758 \\(a,\\,b\\in F\\)\uc5d0 \ub300\ud574 \\(a\\leq b\\) \ub610\ub294 \\(b\\leq a\\)\uc774\ub2e4.<\/li>\n<li>(O2) \\(a\\leq b\\)\uc774\uace0 \\(b\\leq c\\)\uc774\uba74 \\(a\\leq c\\)\uc774\ub2e4.<\/li>\n<li>(O3) \\(a\\leq b\\)\uc774\uace0 \\(b\\leq a\\)\uc774\uba74 \\(a=b\\)\uc774\ub2e4.<\/li>\n<li>(O4) \\(a\\leq b\\)\uc774\uba74 \\(a+c\\leq b+c\\)\uc774\ub2e4.<\/li>\n<li>(O5) \\(a\\leq b\\)\uc774\uace0 \\(0\\leq c\\)\uc774\uba74 \\(ac\\leq bc\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\\(a\\le b\\)\uc774\uba74\uc11c \\(a\\neq b\\)\uc778 \uac83\uc744 \\(a&lt;b\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(a>0\\)\uc77c \ub54c \\(a\\)\ub97c <span class=\"defined\">\uc591\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\(a&lt;0\\)\uc77c \ub54c \\(a\\)\ub97c <span class=\"defined\">\uc74c\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc21c\uc11c\uccb4\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \uc591\uc218, \\(0\\), \uc74c\uc218 \uc911 \uc815\ud655\ud788 \ud558\ub098\uc5d0 \ud574\ub2f9\ud55c\ub2e4.<\/p>\n<p>\uc21c\uc11c\uccb4 \\(F\\)\uc758 \uc6d0\uc18c \\(a\\)\uc758 <span class=\"defined\">\uc808\ub313\uac12<\/span> \\(\\lvert a\\rvert\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\lvert a\\rvert=<br \/>\n\\begin{cases}<br \/>\na &#038; \\text{if }\\;a\\ge0,\\\\[5pt]<br \/>\n-a &#038; \\text{if }\\;a&lt;0.<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.2.<\/span><br \/>\n\uccb4 \\(F\\)\uc758 \uc6d0\uc18c \\(a\\), \\(b\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(0\\cdot a=0\\)<\/li>\n<li>\\(-a=(-1)\\cdot a\\)<\/li>\n<li>\\(-(-a)=a\\)<\/li>\n<li>\\((-a)(-b)=ab\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.3.<\/span><br \/>\n\uc21c\uc11c\uccb4 \\(F\\)\uc758 \uc6d0\uc18c \\(a\\), \\(b\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\lvert ab\\rvert=\\lvert a\\rvert\\lvert b\\rvert\\).<\/li>\n<li>\\(\\lvert a+b\\rvert\\le\\lvert a\\rvert+\\lvert b\\rvert\\). (\uc0bc\uac01\ubd80\ub4f1\uc2dd)<\/li>\n<li>\\(\\lvert\\lvert a\\rvert-\\lvert b\\rvert\\rvert\\le\\lvert a-b\\rvert\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.4.<\/span><br \/>\n\uc21c\uc11c\uccb4\uc758 \uc6d0\uc18c \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(x\\le y\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(x&lt;y+\\varepsilon\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(x=0\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x\\rvert&lt;\\varepsilon\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\ud559\uc5d0\uc11c \uc790\uc8fc \ub2e4\ub8e8\ub294 \ub300\ud45c\uc801\uc778 \uccb4\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uc720\ub9ac\uc218\uccb4 \\(\\mathbb Q\\), \uc2e4\uc218\uccb4 \\(\\mathbb R\\), \ubcf5\uc18c\uc218\uccb4 \\(\\mathbb C\\)\ub294 \ubaa8\ub450 \uccb4\uc774\ub2e4. \ud2b9\ud788 \\(\\mathbb Q\\)\uc640 \\(\\mathbb R\\)\uc740 \uc21c\uc11c\uccb4\uc774\ub2e4.<\/li>\n<li>\\(p\\)\uac00 \uc18c\uc218(prime number)\uc774\uace0 \\(\\mathbb Z_p=\\{0,1,2,\\ldots,p-1\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \ub367\uc148\uacfc \uacf1\uc148\uc744 \ud1b5\uc0c1\uc801\uc778 \uc815\uc218\uc758 \uc5f0\uc0b0\uc758 \uacb0\uacfc\ub97c \\(p\\)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub85c \uc815\uc758\ud558\uc790. \uc608\ub97c \ub4e4\uba74 \\(p=5\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\begin{gathered}<br \/>\n1+0=1,\\quad 1+2=3,\\quad 3+2=0,\\quad 3+4=2,\\quad\\cdots,\\\\<br \/>\n1\\cdot0=0,\\quad 1\\cdot2=2,\\quad 3\\cdot2=1,\\quad 3\\cdot4=2,\\quad\\cdots<br \/>\n\\end{gathered}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\uba74 \\(\\mathbb Z_p\\)\ub294 \uccb4\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb Q\\)\ub294 \uc21c\uc11c\uccb4\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc720\ub9ac\uc218 \uc9d1\ud569\uc5d0\uc11c\ub294 \uc720\ub9ac\uc218 \uc218\uc5f4\uc758 \uadf9\ud55c\uc774 \uc720\ub9ac\uc218\uac00 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ud55c\uacc4\ub97c \uadf9\ubcf5\ud558\uae30 \uc704\ud574 \uc21c\uc11c\uccb4\uc758 \uc644\ube44\uc131\uc774\ub77c\ub294 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n<h3>\uc0c1\ud55c\uacfc \uc644\ube44\uc131<\/h3>\n<p>\uc21c\uc11c\uccb4 \\(F\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(A\\subseteq F\\)\uc640 \uc6d0\uc18c \\(M,\\,m\\in F\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ubaa8\ub4e0 \\(x\\in A\\)\uc5d0 \ub300\ud574 \\(x\\leq M\\)\uc774\uba74 \\(M\\)\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc0c1\uacc4<\/span>(upper bound)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\ubaa8\ub4e0 \\(x\\in A\\)\uc5d0 \ub300\ud574 \\(x\\geq m\\)\uc774\uba74 \\(m\\)\uc744 \\(A\\)\uc758 <span class=\"defined\">\ud558\uacc4<\/span>(lower bound)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uc9d1\ud569 \\(A\\)\uc758 \uc0c1\uacc4\uac00 \uc874\uc7ac\ud560 \ub54c, \u201c\\(A\\)\ub294 <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>\uc774\ub2e4(bounded above)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\uc9d1\ud569 \\(A\\)\uc758 \ud558\uacc4\uac00 \uc874\uc7ac\ud560 \ub54c, \u201c\\(A\\)\ub294 <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>\uc774\ub2e4(bounded below)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\uc9d1\ud569 \\(A\\)\uac00 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc77c \ub54c, \u201c\\(A\\)\ub294 <span class=\"defined\">\uc720\uacc4<\/span>\uc774\ub2e4(bounded)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\uc9d1\ud569 \\(A\\)\uac00 \uc704\ub85c \uc720\uacc4\uc77c \ub54c, \\(A\\)\uc758 \uc0c1\uacc4\uc758 \ucd5c\uc19f\uac12\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc0c1\ud55c<\/span>(supremum) \ub610\ub294 <span class=\"defined\">\ucd5c\uc18c\uc0c1\uacc4<\/span>(least upper bound)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\sup A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc9d1\ud569 \\(A\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uc77c \ub54c, \\(A\\)\uc758 \ud558\uacc4\uc758 \ucd5c\ub313\uac12\uc744 \\(A\\)\uc758 <span class=\"defined\">\ud558\ud55c<\/span>(infimum) \ub610\ub294 <span class=\"defined\">\ucd5c\ub300\ud558\uacc4<\/span>(greatest lower bound)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\inf A\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc989 \\(\\alpha\\in F\\)\uac00 \uc9d1\ud569 \\(A\\)\uc758 \uc0c1\ud55c\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(\\alpha\\)\uac00 \\(A\\)\uc758 \uc0c1\uacc4\uc774\ub2e4.<\/li>\n<li>\\(\\beta\\)\uac00 \\(A\\)\uc758 \uc0c1\uacc4\uc774\uba74, \\(\\alpha\\le\\beta\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\\(\\alpha\\in F\\)\uac00 \uc9d1\ud569 \\(A\\)\uc758 \uc0c1\ud55c\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc73c\ub85c \uc9c4\uc220\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \\(x\\in A\\)\uc5d0 \ub300\ud574 \\(x\\leq\\alpha\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \\(\\alpha-\\varepsilon&lt;x\\)\uc778 \\(x\\in A\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.5.<\/span><br \/>\n\uc0c1\ud55c\uc758 \uc720\uc77c\uc131\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \uc989 \uc9d1\ud569 \\(A\\)\uac00 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \\(\\alpha_1\\)\uacfc \\(\\alpha_2\\)\uac00 \\(A\\)\uc758 \uc0c1\ud55c\uc774\uba74 \\(\\alpha_1=\\alpha_2\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.6.<\/span><br \/>\n\\(A\\subseteq\\mathbb R\\), \\(A\\ne\\varnothing\\)\uc774\uace0 \\(-A=\\{-x\\mid x\\in A\\}\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\alpha\\)\uac00 \\(A\\)\uc758 \uc0c1\ud55c\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(-\\alpha\\)\uac00 \\(-A\\)\uc758 \ud558\ud55c\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.7.<\/span><br \/>\n\\(A\\)\uc640 \\(B\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70 \uc704\ub85c \uc720\uacc4\uc774\uace0, \\(\\alpha=\\sup A\\), \\(\\beta=\\sup B\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(A+B=\\{x+y\\mid x\\in A,\\,y\\in B\\}\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\sup(A+B)=\\alpha+\\beta\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(A\\)\uc640 \\(B\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \\(0\\) \uc774\uc0c1\uc774\uace0 \\(AB=\\{xy\\mid x\\in A,\\,y\\in B\\}\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\sup(AB)=\\alpha\\beta\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<p>\ub2e4\uc74c \uacf5\ub9ac\ub294 \uc2e4\uc218\uacc4\uc640 \uc720\ub9ac\uc218\uacc4\uc758 \ubcf8\uc9c8\uc801\uc778 \ucc28\uc774\ub97c \uc124\uba85\ud558\ub294 \uc9c4\uc220\uc774\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"definition\">\uacf5\ub9ac 1.1. (\uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131)<\/span><\/p>\n<p>\\(A\\)\uac00 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(A\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\uba74, \\(A\\)\uc758 \uc0c1\ud55c\uc774 \\(\\mathbb R\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc77c\ubc18\uc801\uc73c\ub85c \uc21c\uc11c\uccb4 \\(F\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc774 \uc704\ub85c \uc720\uacc4\uc77c \ub54c\ub9c8\ub2e4 \uadf8 \uc0c1\ud55c\uc774 \\(F\\)\uc5d0 \uc874\uc7ac\ud558\uba74, \\(F\\)\uac00 <span class=\"defined\">\uc644\ube44\uc131<\/span>(completeness)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ub9d0\ud558\uace0 \\(F\\)\ub97c <span class=\"defined\">\uc644\ube44\uc21c\uc11c\uccb4<\/span>(complete ordered field)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc644\ube44\uc21c\uc11c\uccb4\ub294 \uc21c\uc11c\uccb4\ub85c\uc11c \ub3d9\ud615\uc774\ub77c\ub294 \uc758\ubbf8\uc5d0\uc11c \uc720\uc77c\ud558\ub2e4\ub294 \uc0ac\uc2e4\uc774 \uc54c\ub824\uc838 \uc788\uc73c\uba70, \uc774 \uad50\uc7ac\uc5d0\uc11c\ub294 \uc774 \uc0ac\uc2e4\uc744 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ud558\ub098\uc758 \uc644\ube44\uc21c\uc11c\uccb4\ub97c \uace0\uc815\ud558\uc5ec <span class=\"defined\">\uc2e4\uc218\uacc4<\/span>(real number system)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\mathbb R\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uacf5\ub9ac 1.1\uc740 \ubc14\ub85c \uc774 \uc644\ube44\uc131\uc744 \uc9c4\uc220\ud55c \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.8.<\/span><br \/>\n\\(B\\)\uac00 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(B\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74, \\(B\\)\uc758 \ud558\ud55c\uc774 \\(\\mathbb R\\)\uc5d0 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc774 \uad50\uc7ac\uc5d0\uc11c\ub294 \uc9d1\ud569 \\(A\\)\uac00 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2d0 \ub54c \\(\\sup A=\\infty\\)\ub85c, \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2d0 \ub54c \\(\\inf A=-\\infty\\)\ub85c \uc4f4\ub2e4. \uc5ec\uae30\uc11c \\(\\infty\\)\uc640 \\(-\\infty\\)\ub294 \uc2e4\uc218\uac00 \uc544\ub2cc \uae30\ud638\uc774\uba70, \\(\\mathbb R\\cup\\{-\\infty,\\infty\\}\\)\ub97c <span class=\"defined\">\ud655\uc7a5\uc2e4\uc218\uacc4<\/span>(extended real number system)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ud655\uc7a5\ub41c \uc2e4\uc218\uac12\uc744 \ud5c8\uc6a9\ud560 \ub54c\uc5d0\ub294 \\(\\sup\\varnothing=-\\infty\\), \\(\\inf\\varnothing=\\infty\\)\ub85c \uc57d\uc18d\ud55c\ub2e4. \ubc18\uba74 \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \uc2e4\uc218\uac12\uc73c\ub85c\ub9cc \ub17c\ud560 \ub54c\uc5d0\ub294 \uacf5\uc9d1\ud569\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.9.<\/span><br \/>\n\ub2e4\uc74c \uc9d1\ud569\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\{2,4,6,8\\}\\)<\/li>\n<li>\uc5f4\ub9b0\uad6c\uac04 \\((1,3)\\)<\/li>\n<li>\\(A=\\left\\{0,\\frac12,\\frac23,\\frac34,\\ldots\\right\\}\\)<\/li>\n<li>\\(B=(1,3)\\setminus\\mathbb Q\\). \uc774 \ud56d\ubaa9\uc740 \ubb38\uc81c 1.20\uc758 \uacb0\uacfc\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>\uc790\uc5f0\uc218, \uc815\uc218, \uc720\ub9ac\uc218<\/h3>\n<p>\\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(N\\)\uc774 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(N\\)\uc744 <span class=\"defined\">\uadc0\ub0a9\uc801 \uc9d1\ud569<\/span>(inductive set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>(IND1) \\(1\\in N\\)\uc774\ub2e4.<\/li>\n<li>(IND2) \\(n\\in N\\)\uc774\uba74 \\(n+1\\in N\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\ub610\ud55c \ubaa8\ub4e0 \uadc0\ub0a9\uc801 \uc9d1\ud569\uc758 \uad50\uc9d1\ud569\uc744 <span class=\"defined\">\uc790\uc5f0\uc218<\/span> \uc9d1\ud569\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\mathbb N\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \\(0\\) \uc774\uc0c1\uc778 \uc815\uc218\ub97c \uc790\uc5f0\uc218\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \uc774 \ub178\ud2b8\uc5d0\uc11c\ub294 \\(1\\) \uc774\uc0c1\uc778 \uc815\uc218\ub97c \uc790\uc5f0\uc218\ub77c\uace0 \ubd80\ub974\uaca0\ub2e4.]<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.10.<\/span><br \/>\n\\(\\phi\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(\\mathbb N\\)\uc778 \uba85\uc81c\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc989 \\(\\phi(n)\\)\uc740 \uc790\uc5f0\uc218 \\(n\\)\uc758 \uac12\uc5d0 \ub530\ub77c \ucc38 \ub610\ub294 \uac70\uc9d3\uc774 \uacb0\uc815\ub418\ub294 \uc9c4\uc220\uc774\ub2e4. \\(\\phi\\)\uac00 \ub2e4\uc74c \ub450 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \uac00\uc815\ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\phi(1)\\)\uc774 \ucc38\uc774\ub2e4.<\/li>\n<li>\\(\\phi(k)\\)\uac00 \ucc38\uc774\uba74 \\(\\phi(k+1)\\)\ub3c4 \ucc38\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\uc774\ub54c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\phi(n)\\)\uc774 \ucc38\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \\(-n\\)\uc744 <span class=\"defined\">\uc74c\uc758 \uc815\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc790\uc5f0\uc218\ub97c <span class=\"defined\">\uc591\uc758 \uc815\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \uc591\uc758 \uc815\uc218\uc640 \uc74c\uc758 \uc815\uc218, \uadf8\ub9ac\uace0 \\(0\\)\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uc815\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ubaa8\ub4e0 \uc815\uc218\uc758 \uc9d1\ud569\uc744 \\(\\mathbb Z\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub450 \uc815\uc218\uc758 \ube44\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \uc218\ub97c <span class=\"defined\">\uc720\ub9ac\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ubb3c\ub860 \\(0\\)\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \ubd84\ubaa8\uac00 \\(0\\)\uc778 \ube44\ub294 \uc0dd\uac01\ud558\uc9c0 \uc54a\ub294\ub2e4. \ubaa8\ub4e0 \uc720\ub9ac\uc218\uc758 \uc9d1\ud569\uc744 \\(\\mathbb Q\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989<br \/>\n\\[<br \/>\n\\mathbb Q=\\left\\{\\frac{m}{k}\\,\\Big\\vert\\,m\\in\\mathbb Z,\\;k\\in\\mathbb Z,\\;k\\neq0\\right\\}.<br \/>\n\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.11.<\/span><br \/>\n\uc790\uc5f0\uc218 \uc9d1\ud569\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.12.<\/span><br \/>\n\\(A\\)\uac00 \\(\\mathbb Z\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \uc704\ub85c \uc720\uacc4\uc77c \ub54c, \\(A\\)\uc758 \uc0c1\ud55c\uc774 \\(\\mathbb Z\\)\uc5d0 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.13.<\/span><br \/>\n\uc9d1\ud569 \\(\\mathbb C=\\mathbb R^2\\)\uc5d0 \ub2e4\uc74c\uacfc \uac19\uc774 \ub367\uc148\uacfc \uacf1\uc148\uc744 \uc815\uc758\ud558\uc790.<br \/>\n\\[<br \/>\n(a,b)+(c,d)=(a+c,b+d),\\quad<br \/>\n(a,b)(c,d)=(ac-bd,ad+bc).<br \/>\n\\]<br \/>\n\uc774 \uc5f0\uc0b0\uc5d0 \ub300\ud558\uc5ec \\(\\mathbb C\\)\uac00 \uccb4\uac00 \ub428\uc744 \ud655\uc778\ud558\uc2dc\uc624. \ub610\ud55c \\(i=(0,1)\\)\ub85c \ub450\uba74 \\(i^2=-1\\)\uc774\uace0 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \\(a+bi\\)\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uc2e4\uc218\uc5f4\uc758 \uadf9\ud55c<\/h3>\n<p>\uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \uc2e4\uc218 \\(L\\)\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 \\(|a_n-L|&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[<br \/>\n\\lim_{n\\to\\infty}a_n=L<br \/>\n\\quad\\text{\ub610\ub294}\\quad<br \/>\na_n\\to L.<br \/>\n\\]<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 1.2. (\uadf9\ud55c\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\uc2e4\uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \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\\(a_n\\to L\\)\uc774\uace0 \\(a_n\\to M\\)\uc774\ub77c\uace0 \ud558\uc790. \\(L\\ne M\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uace0 \\(\\varepsilon=\\frac{|L-M|}{3}>0\\)\uc73c\ub85c \ub450\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|a_n-L|&lt;\\varepsilon,\\quad |a_n-M|&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uac00 \ub3d9\uc2dc\uc5d0 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n|L-M|\\le |L-a_n|+|a_n-M|<br \/>\n&lt;2\\varepsilon=\\frac23|L-M|<br \/>\n\\]<br \/>\n\uac00 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(L=M\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\le M\\)\uc77c \ub54c, \u201c\\(\\{a_n\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc774\ub2e4(bounded above)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc2e4\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\ge m\\)\uc77c \ub54c, \u201c\\(\\{a_n\\}\\)\uc774 \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4(bounded below)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc218\uc5f4\uc774 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc77c \ub54c \u201c\uc218\uc5f4\uc774 \uc720\uacc4\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc989 \uc218\uc5f4\uc774 \uc720\uacc4\ub77c\ub294 \uac83\uc740 \uadf8 \uc218\uc5f4\uc758 \ubaa8\ub4e0 \ud56d\uc744 \uc6d0\uc18c\ub85c \uac16\uace0 \uae38\uc774\uac00 \uc720\ud55c\uc778 \uad6c\uac04\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.14.<\/span><br \/>\n\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774 \uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc0ac\uce59\uacc4\uc0b0 \ubc0f \uc21c\uc11c\uad00\uacc4\uc640 \uad00\ub828\ud558\uc5ec \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ub2e4\uc74c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 1.3. (\uc2e4\uc218\uc5f4 \uadf9\ud55c\uc758 \uc5f0\uc0b0\ubc95\uce59)<\/span><\/p>\n<ul>\n<li>\uc120\ud615\uc131: \\(c\\)\uc640 \\(d\\)\uac00 \uc0c1\uc218\uc774\uace0 \\(a_n\\to A\\), \\(b_n\\to B\\)\uc774\uba74 \\(ca_n+db_n\\to cA+dB\\)\uc774\ub2e4.<\/li>\n<li>\uacf1\uc758 \uadf9\ud55c: \\(a_n\\to A\\), \\(b_n\\to B\\)\uc774\uba74 \\(a_nb_n\\to AB\\)\uc774\ub2e4.<\/li>\n<li>\ubaab\uc758 \uadf9\ud55c: \\(a_n\\to A\\), \\(b_n\\to B\\neq0\\)\uc774\uace0 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(b_n\\neq0\\)\uc774\uba74 \\(a_n\/b_n\\to A\/B\\)\uc774\ub2e4.<\/li>\n<li>\uc21c\uc11c \ubcf4\uc874: \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\leq b_n\\)\uc774\uace0 \\(a_n\\to A\\), \\(b_n\\to B\\)\uc774\uba74 \\(A\\leq B\\)\uc774\ub2e4.<\/li>\n<li>\uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac: \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\leq b_n\\leq c_n\\)\uc774\uace0 \\(a_n\\to L\\), \\(c_n\\to L\\)\uc774\uba74 \\(b_n\\to L\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \uc120\ud615\uc131\uc744 \ubcf4\uc774\uc790. \\(\\varepsilon>0\\)\uc774\ub77c\uace0 \ud558\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|a_n-A|&lt;\\frac{\\varepsilon}{2(|c|+1)},\\quad<br \/>\n|b_n-B|&lt;\\frac{\\varepsilon}{2(|d|+1)}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n|ca_n+db_n-(cA+dB)|<br \/>\n\\le |c|\\,|a_n-A|+|d|\\,|b_n-B|<br \/>\n&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(ca_n+db_n\\to cA+dB\\)\uc774\ub2e4.<\/p>\n<p>\uacf1\uc758 \uadf9\ud55c\uc744 \ubcf4\uc774\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(|a_n-A|&lt;1\\)\uc774\ubbc0\ub85c \\(|a_n|\\le |A|+1\\)\uc774\ub2e4. \ub610\ud55c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|b_n-B|&lt;\\frac{\\varepsilon}{2(|A|+1)},\\quad<br \/>\n|a_n-A|&lt;\\frac{\\varepsilon}{2(|B|+1)}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n|a_nb_n-AB|<br \/>\n\\le |a_n|\\,|b_n-B|+|B|\\,|a_n-A|<br \/>\n&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(a_nb_n\\to AB\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(b_n\\to B\\ne0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc774\uba74<br \/>\n\\[<br \/>\n|b_n-B|&lt;<br \/>\n\\min\\left\\{\\frac{|B|}{2},\\,\\frac{\\varepsilon|B|^2}{2}\\right\\}.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(|b_n|>|B|\/2\\)\uc774\uace0<br \/>\n\\[<br \/>\n\\left|\\frac1{b_n}-\\frac1B\\right|<br \/>\n=\\frac{|b_n-B|}{|b_n|\\,|B|}<br \/>\n\\le\\frac2{|B|^2}|b_n-B|<br \/>\n&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(1\/b_n\\to1\/B\\)\uc774\uace0, \uacf1\uc758 \uadf9\ud55c\uc744 \uc801\uc6a9\ud558\uba74 \\(a_n\/b_n\\to A\/B\\)\uc774\ub2e4.<\/p>\n<p>\uc21c\uc11c \ubcf4\uc874\uc744 \ubcf4\uc774\uc790. \ub9cc\uc57d \\(A>B\\)\uc774\uba74 \\(\\varepsilon=(A-B)\/3\\)\uc73c\ub85c \ub458 \uc218 \uc788\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n>A-\\varepsilon>B+\\varepsilon>b_n<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(a_n\\le b_n\\)\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(A\\le B\\)\uc774\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(a_n\\le b_n\\le c_n\\), \\(a_n\\to L\\), \\(c_n\\to L\\)\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc774\uba74<br \/>\n\\[<br \/>\nL-\\varepsilon&lt;a_n\\le b_n\\le c_n&lt;L+\\varepsilon<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(|b_n-L|&lt;\\varepsilon\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(b_n\\to L\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.15.<\/span><br \/>\n\\(a_n\\to A\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(|a_n|\\to|A|\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\ne0\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\)\uacfc \\(A\\)\uc758 \ubd80\ud638\uac00 \uac19\uace0 \\(|a_n|>|A|\/2\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \uc5b4\ub5a0\ud55c \uc2e4\uc218\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c, \u201c\\(\\{a_n\\}\\)\uc774 <span class=\"defined\">\ubc1c\uc0b0<\/span>\ud55c\ub2e4(diverge)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ubc1c\uc0b0\ud558\ub294 \uc591\uc0c1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ubd84\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \\(M>0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\)\uc77c \ub54c \\(a_n>M\\)\uc774\uba74 \u201c\\(\\{a_n\\}\\)\uc774 <span class=\"defined\">\uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0<\/span>\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \\(\\displaystyle\\lim_{n\\to\\infty}a_n=\\infty\\) \ub610\ub294 \\(a_n\\to\\infty\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(M>0\\)\uc5d0 \ub300\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\)\uc77c \ub54c \\(a_n&lt;-M\\)\uc774\uba74 \u201c\\(\\{a_n\\}\\)\uc774 <span class=\"defined\">\uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0<\/span>\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \\(\\displaystyle\\lim_{n\\to\\infty}a_n=-\\infty\\) \ub610\ub294 \\(a_n\\to-\\infty\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc218\uc5f4\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uace0, \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc9c0 \uc54a\uace0, \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc9c0\ub3c4 \uc54a\uc744 \ub54c, \u201c\uc218\uc5f4\uc774 <span class=\"defined\">\uc9c4\ub3d9<\/span>\ud55c\ub2e4(oscillate)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ul>\n<h3>\uc644\ube44\uc131\uacfc \uad00\ub828 \uc815\ub9ac\ub4e4<\/h3>\n<p>\uc720\uacc4\uc778 \uc218\uc5f4\uc774 \ud56d\uc0c1 \uc218\ub834\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ub098 \ub2e8\uc870\uc774\uba74\uc11c \uc720\uacc4\uc778 \uc218\uc5f4\uc740 \ubc18\ub4dc\uc2dc \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\le a_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\{a_n\\}\\)\uc744 <span class=\"defined\">\ub2e8\uc870\uc99d\uac00<\/span>\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n&lt;a_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\{a_n\\}\\)\uc744 <span class=\"defined\">\uc21c\uc99d\uac00<\/span>\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\ge a_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\{a_n\\}\\)\uc744 <span class=\"defined\">\ub2e8\uc870\uac10\uc18c<\/span>\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n>a_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\{a_n\\}\\)\uc744 <span class=\"defined\">\uc21c\uac10\uc18c<\/span>\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uacfc \ub2e8\uc870\uac10\uc18c\ud558\ub294 \uc218\uc5f4\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\ub2e8\uc870\uc218\uc5f4<\/span>(monotone sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 1.4. (\ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p>\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc218\uc5f4\uc774\uace0 \uc720\uacc4\uc774\uba74, \\(\\{a_n\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(A=\\{a_n\\mid n\\in\\mathbb N\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc0c1\ud55c \\(\\alpha=\\sup A\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\\(\\varepsilon>0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\alpha-\\varepsilon&lt;a_N\\le\\alpha\\)\uc774\ub2e4. \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\ubbc0\ub85c \\(n>N\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n\\alpha-\\varepsilon&lt;a_N\\leq a_n\\leq\\alpha,<br \/>\n\\]<br \/>\n\uc989 \\(|a_n-\\alpha|&lt;\\varepsilon\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(a_n\\to\\alpha\\)\uc774\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uba74 \\(b_n=-a_n\\)\uc73c\ub85c \uc815\uc758\ub41c \\(\\{b_n\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \uc720\uacc4\uc778 \uc218\uc5f4\uc774\ubbc0\ub85c \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\{a_n\\}\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec <span class=\"defined\">\uc790\uc5f0\uc0c1\uc218<\/span> \\(e\\)\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc989<br \/>\n\\[<br \/>\ne_n=\\left(1+\\frac1n\\right)^n<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \uc815\uc758\ub41c \uc218\uc5f4 \\(\\{e_n\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc218\ub834\ud55c\ub2e4. \uc774 \uadf9\ud55c\uc744 \\(e\\)\ub85c \uc815\uc758\ud55c\ub2e4. \uc989<br \/>\n\\[<br \/>\ne=\\lim_{n\\to\\infty}\\left(1+\\frac1n\\right)^n<br \/>\n=2.718281828459045\\cdots.<br \/>\n\\]<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.16.<\/span><br \/>\n\\(e_n=\\left(1+\\frac1n\\right)^n\\)\uc774\ub77c\uace0 \uc815\uc758\ub41c \uc218\uc5f4 \\(\\{e_n\\}\\)\uc774 \ub2e8\uc870\uc774\uace0 \uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \uae38\uc774\uac00 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ub294 \ub2eb\ud78c \ucd95\uc18c\uad6c\uac04\uc5f4\uc758 \uad50\uc9d1\ud569\uc774 \ub2e8 \ud558\ub098\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4\ub2e4\ub294 \uc131\uc9c8\uc744 \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 1.5. (\ucd95\uc18c\uad6c\uac04 \uc815\ub9ac)<\/span><\/p>\n<p>\ub2eb\ud78c\uad6c\uac04\uc758 \uc218\uc5f4 \\(\\{[a_n,b_n]\\}\\)\uc774 \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\([a_{n+1},b_{n+1}]\\subseteq[a_n,b_n]\\)\uc774\ub2e4.<\/li>\n<li>\\(\\displaystyle\\lim_{n\\to\\infty}(b_n-a_n)=0\\).<\/li>\n<\/ul>\n<p>\uadf8\ub7ec\uba74 \\(\\displaystyle\\bigcap_{n=1}^{\\infty}[a_n,b_n]\\)\uc740 \uc815\ud655\ud788 \ud55c \uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uad6c\uac04\ub4e4\uc774 \ucd95\uc18c\ud558\ubbc0\ub85c \\(a_n\\le a_{n+1}\\le b_{n+1}\\le b_n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\{a_n\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \\(b_1\\)\uc744 \uc0c1\uacc4\ub85c \uac00\uc9c0\uba70, \\(\\{b_n\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \\(a_1\\)\uc744 \ud558\uacc4\ub85c \uac00\uc9c4\ub2e4. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \ub530\ub77c<br \/>\n\\[<br \/>\na_n\\to A,\\quad b_n\\to B<br \/>\n\\]<br \/>\n\uc778 \uc2e4\uc218 \\(A,B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(b_n-a_n\\to0\\)\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc5f0\uc0b0\ubc95\uce59\uc5d0 \uc758\ud574 \\(B-A=0\\), \uc989 \\(A=B\\)\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uace0\uc815\ub41c \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(m\\ge n\\)\uc774\uba74 \\(a_n\\le a_m\\le b_m\\le b_n\\)\uc774\ub2e4. \\(m\\to\\infty\\)\ub85c \ubcf4\ub0b4\uace0 \uc21c\uc11c \ubcf4\uc874\uc744 \uc801\uc6a9\ud558\uba74 \\(a_n\\le L\\le b_n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(L\\)\uc740 \ubaa8\ub4e0 \\([a_n,b_n]\\)\uc5d0 \uc18d\ud55c\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \\(x\\)\uac00 \ubaa8\ub4e0 \\([a_n,b_n]\\)\uc5d0 \uc18d\ud558\uba74 \\(a_n\\le x\\le b_n\\)\uc774\ub2e4. \\(n\\to\\infty\\)\ub85c \ubcf4\ub0b4\uba74 \\(L\\le x\\le L\\)\uc774\ubbc0\ub85c \\(x=L\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uad50\uc9d1\ud569\uc740 \uc815\ud655\ud788 \\(\\{L\\}\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\ub294 \uc2e4\uc218\ub97c \uc2ed\uc9c4\ubc95 \uc804\uac1c\ub85c \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc744 \uc815\ub2f9\ud654\ud558\ub294 \ub370 \uc0ac\uc6a9\ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc6d0\uc8fc\uc728\uc758 \uc2ed\uc9c4\ubc95 \ud45c\ud604 \\(\\pi=3.14159\\cdots\\)\uc758 \uac12\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc740 \uad6c\uac04<br \/>\n\\[<br \/>\n[3,4],\\ [3.1,3.2],\\ [3.14,3.15],\\ \\cdots<br \/>\n\\]<br \/>\n\uc758 \uad50\uc9d1\ud569\uc5d0 \ub2e8 \ud558\ub098\uc758 \uc6d0\uc18c\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \uc758\ud558\uc5ec \ubcf4\uc7a5\ub41c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.17.<\/span><br \/>\n\ucd95\uc18c\uad6c\uac04 \uc815\ub9ac\uc5d0\uc11c \ub2eb\ud78c\uad6c\uac04 \ub300\uc2e0 \uc5f4\ub9b0\uad6c\uac04\uc744 \uc0ac\uc6a9\ud558\uba74 \uacb0\ub860\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\uc74c\uc744 \uc608\ub97c \ub4e4\uc5b4 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc218\uc5f4 \\(\\{a_n\\}\\)\uacfc \uc99d\uac00\ud558\ub294 \uc790\uc5f0\uc218\uc5f4 \\(n_1&lt;n_2&lt;\\cdots\\)\uc5d0 \ub300\ud558\uc5ec \\(\\{a_{n_k}\\}\\)\ub97c \\(\\{a_n\\}\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc218\uc5f4<\/span>(subsequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ubd80\ubd84\uc218\uc5f4\uc740 \uc6d0\ub798 \uc218\uc5f4\uc5d0\uc11c \ud56d\uc758 \uc21c\uc11c\ub97c \uc720\uc9c0\ud55c \ucc44 \uc77c\ubd80 \ud56d\uc744 \uace8\ub77c \uc5bb\ub294 \uc218\uc5f4\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 1.6. (\ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4)<\/span><\/p>\n<p>\uc720\uacc4\uc778 \uc218\uc5f4\uc740 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(\\{a_n\\}\\)\uc774 \uc720\uacc4\uc778 \uc218\uc5f4\uc774\uace0 \ubaa8\ub4e0 \ud56d\uc774 \ub2eb\ud78c\uad6c\uac04 \\([a,b]\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud558\uc790. \\([a,b]\\)\ub97c \uc774\ub4f1\ubd84\ud55c \ub450 \ub2eb\ud78c\uad6c\uac04 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uc5d0\ub294 \\(\\{a_n\\}\\)\uc758 \ud56d\uc774 \ubb34\ud55c\ud788 \ub9ce\uc774 \uc18d\ud55c\ub2e4. \uadf8\ub7ec\ud55c \uad6c\uac04 \ud558\ub098\ub97c \\(I_1=[c_1,d_1]\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud55c\ub2e4. \uc989 \\(I_k=[c_k,d_k]\\)\uac00 \uc815\ud574\uc84c\uc73c\uba74 \uc774\ub97c \uc774\ub4f1\ubd84\ud55c \ub450 \ub2eb\ud78c\uad6c\uac04 \uc911 \\(\\{a_n\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \ud3ec\ud568\ud558\ub294 \ud558\ub098\ub97c \\(I_{k+1}=[c_{k+1},d_{k+1}]\\)\ub85c \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\nI_{k+1}\\subseteq I_k,\\quad<br \/>\nd_k-c_k=\\frac{b-a}{2^k}.<br \/>\n\\]<br \/>\n\ubb38\uc81c 1.11\uc758 \uacb0\uacfc\uc640 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c \uc5bb\ub294 \\(2^k\\ge k+1\\)\uc744 \uc774\uc6a9\ud558\uba74 \\((b-a)\/2^k\\to0\\)\uc784\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \ucd95\uc18c\uad6c\uac04 \uc815\ub9ac\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n\\bigcap_{k=1}^{\\infty}I_k=\\{L\\}<br \/>\n\\]<br \/>\n\uc778 \uc2e4\uc218 \\(L\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uac01 \\(I_k\\)\uc5d0\ub294 \uc218\uc5f4\uc758 \ud56d\uc774 \ubb34\ud55c\ud788 \ub9ce\uc774 \uc18d\ud558\ubbc0\ub85c \\(n_1&lt;n_2&lt;\\cdots\\)\uac00 \ub418\ub3c4\ub85d \\(a_{n_k}\\in I_k\\)\uc778 \uc9c0\uc218 \\(n_k\\)\ub97c \ucc28\ub840\ub85c \ud0dd\ud560 \uc218 \uc788\ub2e4. \\(L\\in I_k\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n|a_{n_k}-L|\\le d_k-c_k\\longrightarrow0.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\{a_{n_k}\\}\\)\ub294 \\(L\\)\ub85c \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc5b4\ub5a4 \uc218\uc5f4\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc124\uba85\ud558\ub824\uba74 \uadf9\ud55c\uac12\uc744 \uc54c\uc544\uc57c \ud55c\ub2e4. \uadf8\ub7ec\ub098 \uadf9\ud55c\uac12\uc744 \uc54c\uc9c0 \ubabb\ud55c \uc0c1\ud0dc\uc5d0\uc11c\ub3c4 \uc218\uc5f4\uc758 \uc218\ub834\uc744 \uc124\uba85\ud560 \uc218 \uc788\ub294 \ubc29\ubc95\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc218\uc5f4 \\(\\{a_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 \\(|a_m-a_n|&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 1.7. (\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \ucf54\uc2dc \ud310\uc815\ubc95)<\/span><\/p>\n<p>\uc2e4\uc218\uc5f4\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ucf54\uc2dc \uc218\uc5f4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(a_n\\to L\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(m,n\\)\uc774\uba74<br \/>\n\\[<br \/>\n|a_m-L|&lt;\\varepsilon\/2,\\quad<br \/>\n|a_n-L|&lt;\\varepsilon\/2<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n|a_m-a_n|<br \/>\n\\le |a_m-L|+|a_n-L|<br \/>\n&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\{a_n\\}\\)\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\{a_n\\}\\)\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(\\varepsilon=1\\)\uc744 \uc801\uc6a9\ud558\uba74 \uc5b4\ub5a4 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m,n>N\\)\uc77c \ub54c \\(|a_m-a_n|&lt;1\\)\uc774\ub2e4. \ud2b9\ud788 \\(n>N\\)\uc774\uba74 \\(|a_n-a_{N+1}|&lt;1\\)\uc774\ubbc0\ub85c \uaf2c\ub9ac \ubd80\ubd84\uc740 \uc720\uacc4\uc774\uace0, \ucc98\uc74c \uc720\ud55c \uac1c\uc758 \ud56d\uae4c\uc9c0 \ud569\uce58\uba74 \uc218\uc5f4 \uc804\uccb4\uac00 \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac\uc5d0 \ub530\ub77c \\(a_{n_k}\\to L\\)\uc778 \ubd80\ubd84\uc218\uc5f4\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc744 \ud0dd\ud558\uc790. \ucf54\uc2dc \uc870\uac74\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m,n>N\\)\uc774\uba74 \\(|a_m-a_n|&lt;\\varepsilon\/2\\)\uc774\ub2e4. \ub610\ud55c \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(n_k>N\\)\uc774\uace0 \\(|a_{n_k}-L|&lt;\\varepsilon\/2\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(n>N\\)\uc774\uba74<br \/>\n\\[<br \/>\n|a_n-L|<br \/>\n\\le |a_n-a_{n_k}|+|a_{n_k}-L|<br \/>\n&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(a_n\\to L\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc815\ub9ac 1.7\uc740 \uc2e4\uc218\uacc4\uc640 \uc720\ub9ac\uc218\uacc4\ub97c \uad6c\ubd84\ud558\uac8c \ud574\uc8fc\ub294 \uc911\uc694\ud55c \uc131\uc9c8\uc774\ub2e4. \uc989 \\(\\mathbb Q\\)\uc5d0\uc11c\ub294 \ucf54\uc2dc \uc218\uc5f4\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[<br \/>\ne_n=\\left(1+\\frac1n\\right)^n<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \uc815\uc758\ub41c \uc218\uc5f4 \\(\\{e_n\\}\\)\uc740 \ubaa8\ub4e0 \ud56d\uc774 \uc720\ub9ac\uc218\uc774\uace0 \ucf54\uc2dc \uc218\uc5f4\uc774\uc9c0\ub9cc \uadf9\ud55c\uc740 \uc720\ub9ac\uc218\uac00 \uc544\ub2c8\ub2e4. (\ub4a4\uc5d0\uc11c \ub2e4\ub8f0 \\(e\\)\uc758 \ubb34\ub9ac\uc218\uc131 \ubb38\uc81c \ucc38\uc870.)<!-- TODO: problem:irrationalityofe\uc758 \ucd5c\uc885 \ubb38\uc81c \ubc88\ud638 \ubc0f \uc628\ub77c\uc778 \ub9c1\ud06c \ud655\uc778 --><\/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\ud55c \ub4a4\uc5d0\ub294, \ubaa8\ub4e0 \ucf54\uc2dc \uc218\uc5f4\uc774 \uadf8 \uacf5\uac04\uc758 \uc810\uc73c\ub85c \uc218\ub834\ud558\ub294 \uac70\ub9ac\uacf5\uac04\uc744 \uc644\ube44\ub77c\uace0 \ubd80\ub974\uac8c \ub41c\ub2e4. (\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\uc758 \uc218\ub834\uc744 \ub2e4\ub8e8\ub294 \uc808 \ucc38\uc870.)<!-- TODO: section:convergenceinmetricspace\uc758 \ucd5c\uc885 \uc808 \ubc88\ud638 \ud655\uc778 --><br \/>\n\uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc815\ub9ac 1.7\uc740 \uc2e4\uc218\uacc4 \\(\\mathbb R\\)\uc774 \uc644\ube44\uc778 \uac70\ub9ac\uacf5\uac04\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.18.<\/span><br \/>\n\\(\\{a_n\\}\\)\uc774 \uc815\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\(\\{a_n\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc815\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\{a_n\\}\\)\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\uba74 \\(\\{a_n\\}\\)\uc774 \uc218\ub834\ud558\uace0, \uadf8 \uadf9\ud55c\uc774 \uc815\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uba87 \uac00\uc9c0 \uc720\uc6a9\ud55c \uc131\uc9c8<\/h3>\n<p>\ud574\uc11d\ud559\uc758 \uc774\ub860\uc744 \uc804\uac1c\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uba87 \uac00\uc9c0 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 1.8. (\uc544\ub974\ud0a4\uba54\ub370\uc2a4 \uc131\uc9c8)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc591\uc218 \\(x,\\,y\\)\uc5d0 \ub300\ud574 \\(nx>y\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uadf8\ub7ec\ud55c \\(n\\)\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc9d1\ud569<br \/>\n\\[<br \/>\nE=\\{nx\\mid n\\in\\mathbb N\\}<br \/>\n\\]<br \/>\n\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c, \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131 \uacf5\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c \\(\\alpha\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\alpha-x\\)\ub294 \\(E\\)\uc758 \uc0c1\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c, \\(mx>\\alpha-x\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \\((m+1)x>\\alpha\\)\uc778\ub370, \\((m+1)x\\in E\\)\uc774\ubbc0\ub85c, \uc774\uac83\uc740 \\(\\alpha\\)\uac00 \\(E\\)\uc758 \uc0c1\ud55c\uc774\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc544\ub974\ud0a4\uba54\ub370\uc2a4 \uc131\uc9c8\ub85c\ubd80\ud130 \ub2e4\uc74c \ub450 \uc131\uc9c8\uc774 \ubc14\ub85c \ub530\ub77c\uc628\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\ub530\ub984\uc815\ub9ac 1.9. (\uc544\ub974\ud0a4\uba54\ub370\uc2a4 \uc131\uc9c8\uc758 \uadc0\uacb0)<\/span><\/p>\n<ul>\n<li>\uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \\(\\frac1n&lt;\\varepsilon\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud574 \\(n\\leq x&lt;n+1\\)\uc778 \uc815\uc218 \\(n\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(n=\\lfloor x\\rfloor\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uccab \ubc88\uc9f8 \uc131\uc9c8\uc740 \uc815\ub9ac 1.8\uc5d0\uc11c \\(x=\\varepsilon\\), \\(y=1\\)\ub85c \ub450\uba74 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub450 \ubc88\uc9f8 \uc131\uc9c8\uc744 \ubcf4\uc774\uc790. \\(x\\in\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nA=\\{k\\in\\mathbb Z\\mid k\\le x\\}<br \/>\n\\]<br \/>\n\ub85c \ub450\uc790. \uc544\ub974\ud0a4\uba54\ub370\uc2a4 \uc131\uc9c8\uc5d0 \uc758\ud574 \\(m>|x|\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\ubbc0\ub85c \\(-m\\in A\\)\uc774\uace0 \\(A\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \ub610\ud55c \\(x\\)\uac00 \\(A\\)\uc758 \uc0c1\uacc4\uc774\ub2e4. \ubb38\uc81c 1.12\ub97c \uc801\uc6a9\ud558\uba74 \\(n=\\sup A\\)\uc778 \uc815\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(x\\)\uac00 \\(A\\)\uc758 \uc0c1\uacc4\uc774\ubbc0\ub85c \\(n\\le x\\)\uc774\uace0, \ub9cc\uc57d \\(x\\ge n+1\\)\uc774\uba74 \\(n+1\\in A\\)\uac00 \ub418\uc5b4 \\(n=\\sup A\\)\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(n\\le x&lt;n+1\\)\uc774\ub2e4.<\/p>\n<p>\uc720\uc77c\uc131\ub3c4 \ubc14\ub85c \ub530\ub978\ub2e4. \uc2e4\uc81c\ub85c \uc815\uc218 \\(k&lt;\\ell\\)\uc774\uba74 \\(\\ell-k\\)\uac00 \uc591\uc758 \uc815\uc218\uc774\ubbc0\ub85c \\(\\ell-k\\ge1\\), \uc989 \\(k+1\\le\\ell\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub450 \uc815\uc218 \\(k,\\ell\\)\uac00 \ubaa8\ub450<br \/>\n\\[<br \/>\nk\\le x&lt;k+1,\\quad<br \/>\n\\ell\\le x&lt;\\ell+1<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0ac \uc218 \uc5c6\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc55e\uc73c\ub85c \\(a>0\\), \\(n\\in\\mathbb N\\)\uc77c \ub54c \\(x^n=a\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc591\uc218 \\(x\\)\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uc774 \uc0ac\uc2e4\uc740 \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131\uc73c\ub85c\ubd80\ud130 \uc99d\uba85\ud560 \uc218 \uc788\uc73c\uba70, \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \uc774 \\(x\\)\ub97c \\(a^{1\/n}\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.19.<\/span><br \/>\n\uc784\uc758\uc758 \uc2e4\uc218 \\(a&lt;b\\)\uc5d0 \ub300\ud574 \\(a&lt;r&lt;b\\)\uc778 \uc720\ub9ac\uc218 \\(r\\)\uc774 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624. (\uc774 \uc131\uc9c8\uc744 <span class=\"defined\">\uc720\ub9ac\uc218\uc758 \uc870\ubc00\uc131<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.20.<\/span><br \/>\n\uc784\uc758\uc758 \uc2e4\uc218 \\(a&lt;b\\)\uc5d0 \ub300\ud574 \\(a&lt;s&lt;b\\)\uc778 \ubb34\ub9ac\uc218 \\(s\\)\uac00 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624. (\uc774 \uc131\uc9c8\uc744 <span class=\"defined\">\ubb34\ub9ac\uc218\uc758 \uc870\ubc00\uc131<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\sqrt2\\notin\\mathbb Q\\)\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.21.<\/span><br \/>\n\\(x\\ge-1\\)\uc774\uace0 \\(r\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<br \/>\n\\[<br \/>\n(1+x)^r\\ge1+rx.<br \/>\n\\]<br \/>\n\uc774 \ubd80\ub4f1\uc2dd\uc744 <span class=\"defined\">\ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd<\/span>(Bernoulli&#8217;s inequality)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc5d0\uc11c \\(r\\)\uc758 \ubc94\uc704\ub97c \uc2e4\uc218\ub85c \ud655\uc7a5\ud55c \ud615\ud0dc\ub294 \ub4a4\uc758 \uc77c\ubc18\ud654\ub41c \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd \ubb38\uc81c\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.<!-- TODO: bernoulliinequalitygeneral\uc758 \ucd5c\uc885 \ubb38\uc81c \ubc88\ud638 \ubc0f \uc628\ub77c\uc778 \ub9c1\ud06c \ud655\uc778 --><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.22.<\/span><br \/>\n\uc2e4\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec \uc218\uc5f4 \\(\\{r^n\\}\\)\uc774 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(r\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud558\uc2dc\uc624. \ub610\ud55c \\(\\{r^n\\}\\)\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.23.<\/span><br \/>\n\\(a>0\\)\uc77c \ub54c \uc218\uc5f4 \\(\\left\\{a^{1\/n}\\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uace0, \uc774 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624. (\ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc640 \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.24.<\/span><br \/>\n\\(a>0\\)\uc77c \ub54c \uc218\uc5f4 \\(\\left\\{\\frac{a^n}{n!}\\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uace0, \uc774 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624. (\ubb38\uc81c 1.22\uc758 \uacb0\uacfc\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.25.<\/span><br \/>\n\uc218\uc5f4 \\(\\left\\{n^{1\/n}\\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uace0, \uc774 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624. (\\(x_n=n^{1\/n}-1\\)\ub85c \ub450\uace0 \uc774\ud56d\uc815\ub9ac\uc5d0\uc11c \uc774\ucc28\ud56d\uae4c\uc9c0 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\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 class=\"contentboxthis\"><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><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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\uc218\uacc4\ub97c \uc644\ube44\uc21c\uc11c\uccb4\ub85c \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc21c\uc11c\uccb4 \uc9d1\ud569 \\(F\\)\uc5d0 \ub367\uc148\uacfc \uacf1\uc148\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uc774 \uc5f0\uc0b0\uc774 \ub2e4\uc74c \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(F\\)\ub97c \uccb4(field)\ub77c\uace0 \ubd80\ub978\ub2e4. (F1) \ub367\uc148: \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0 0\uc774 \uc874\uc7ac\ud558\uba70, \uc784\uc758\uc758 \uc6d0\uc18c\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc874\uc7ac\ud55c\ub2e4. (F2) \uacf1\uc148: \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0 1\uc774 \uc874\uc7ac\ud558\uba70, 0\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc6d0\uc18c\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774 \uc874\uc7ac\ud55c\ub2e4. \ud2b9\ud788 \\(1\\ne 0\\)\uc774\ub2e4. (F3) \ubd84\ubc30\ubc95\uce59:&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":101,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9473","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9473","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=9473"}],"version-history":[{"count":12,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9473\/revisions"}],"predecessor-version":[{"id":10107,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9473\/revisions\/10107"}],"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=9473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}