{"id":2837,"date":"2019-03-05T23:45:38","date_gmt":"2019-03-05T14:45:38","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2837"},"modified":"2019-12-02T22:50:39","modified_gmt":"2019-12-02T13:50:39","slug":"calculus-limit-of-a-sequence-introduction","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-limit-of-a-sequence-introduction\/","title":{"rendered":"\uc218\uc5f4\uc758 \uadf9\ud55c"},"content":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0 \uadf9\ud55c\uacfc \uad00\ub828\ub41c \uae30\ubcf8\uc801\uc778 \uc131\uc9c8\uc744 \uc99d\uba85\ud55c\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#definition\">\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758<\/a><\/li>\n<li><a href=\"#algebraicproperty\">\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"#analyticproperty\">\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"#divergent\">\ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uc9d1\ud569\uacfc \uc2e4\uc218 (<a href=\"\/blog\/articles\/calculus-sets-and-the-real-number-system\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"definition\"><\/a><\/p>\n<h3>\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758<\/h3>\n<p>\uc801\ub2f9\ud55c \uc815\uc218 \\(n_0\\)\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\uc5ed\uc744 \\(\\left\\{ n \\in \\mathbb{Z} \\,\\vert\\, n \\ge n_0 \\right\\}\\) \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ud568\uc218\ub97c <span class=\"defined\">\uc810\uc5f4<\/span>(sequence)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc778 \uc810\uc5f4\uc744 <span class=\"defined\">\uc2e4\uc218\uc5f4<\/span>(real sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\ubbf8\uc801\ubd84\ud559\uc744 \uacf5\ubd80\ud558\ub294 \ub3d9\uc548\uc5d0\ub294 \uc2e4\uc218\uc5f4\uc744 \uac04\ub2e8\ud788 <span class=\"defined\">\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub85c \ud558\uc790.] \uc608\ucee8\ub300<br \/>\n\\[x_n = 3^n\\tag{1}\\]<br \/>\n\uc740 \uc815\uc218 \\(n\\)\uc744 \\(3^n\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \uc774 \uc218\uc5f4\uc774 \uc815\uc758\uc5ed\uc740 \ubb38\ub9e5\uc5d0 \ub530\ub77c \\(\\mathbb{N}\\)\uc774\ub77c\uace0 \ud560 \uc218\ub3c4 \uc788\uace0 \\(0\\) \uc774\uc0c1\uc778 \uc815\uc218\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud560 \uc218\ub3c4 \uc788\ub2e4. (1)\uacfc \uac19\uc774 \ud568\uc218\uc758 \uc774\ub984\uc774 \\(x\\)\uc77c \ub54c, \uc774 \uc218\uc5f4\uc744 \\(\\left\\{ x_n \\right\\}\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud55c\ud3b8<br \/>\n\\[y_n = \\frac{n+1}{ n(n-1)(n-4)}\\tag{2}\\]<br \/>\n\uc740 \\(n=0\\) \ub610\ub294 \\(n=1\\) \ub610\ub294 \\(n=4\\)\uc77c \ub54c \uc704 \uc2dd\uc758 \uac12\uc774 \uc815\uc758\ub418\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \uc704\uc640 \uac19\uc774 \uc815\uc758\ub41c \uc218\uc5f4 \\(\\left\\{ y_n \\right\\}\\)\uc758 \uc815\uc758\uc5ed\uc740 \\(4\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc758 \ubaa8\uc784\uc774\ub2e4.<\/p>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0\uc11c \\(n\\)\uc774 \uc815\ud574\uc9c8 \ub54c\ub9c8\ub2e4 \\(x_n\\)\ub3c4 \ud558\ub098\uc758 \uac12\uc744 \uac16\ub294\ub370, \uc774\ub54c \\(x_n\\)\uc744 \\(\\left\\{ x_n \\right\\}\\)\uc758 <span class=\"defined\">\ud56d<\/span>(term)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud56d \uc911\uc5d0\uc11c \ucca8\uc790\uac00 \uac00\uc7a5 \uc791\uc740 \ud56d\uc744 <span class=\"defined\">\uccab\uc9f8\ud56d<\/span>(initial term)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ucee8\ub300 (2)\uc640 \uac19\uc774 \uc815\uc758\ub41c \uc218\uc5f4\uc758 \uccab\uc9f8\ud56d\uc740 \\(y_2\\)\uc774\ub2e4.<\/p>\n<p>\uc720\ud55c \uac1c\uc758 \ud56d\uc740 \uadf9\ud55c\uac12\uc5d0 \uc601\ud5a5\uc744 \ubbf8\uce58\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \ub17c\ud560 \ub54c \uc218\uc5f4\uc758 \uccab\uc9f8\ud56d\uc774 \uc5b4\ub514\uc11c\ubd80\ud130 \uc2dc\uc791\ud558\ub294\uc9c0\ub294 \ud06c\uac8c \uc911\uc694\ud558\uc9c0 \uc54a\ub2e4. [\ub2e8, \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \uad6c\ud560 \ub54c\uc5d0\ub294 \uc218\uc5f4\uc758 \uccab\uc9f8\ud56d\uc774 \uc5b4\ub514\uc11c\ubd80\ud130 \uc2dc\uc791\ud558\ub294\uc9c0\ub3c4 \uc911\uc694\ud558\ub2e4.]<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc218\uc5f4\uc758 \uadf9\ud55c)<\/span><\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert x_n &#8211; L \\right\\rvert < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \u2018\\(n\\)\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c8 \ub54c \\(x_n\\)\uc740 \\(L\\)\uc5d0 <span class=\"defined\">\uc218\ub834<\/span>\ud55c\ub2e4\u2019 \ub610\ub294 \u2018\\(\\left\\{ x_n \\right\\}\\)\uc758 <span class=\"defined\">\uadf9\ud55c<\/span>\uc740 \\(L\\)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294<br \/>\n\\[x_n \\,\\,\\rightarrow\\,\\, L \\tag{3}\\]<br \/>\n\ub610\ub294<br \/>\n\\[\\lim_{n\\to\\infty} x_n = L \\tag{4}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc5b4\ub5a0\ud55c \uac12\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\ub2e4\uc74c \ud45c\ud604\uc740 \ubaa8\ub450 \uac19\uc740 \ub73b\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(n\\)\uc774 \ubb34\ud55c\ud788 \ucee4\uc9c8 \ub54c \\(x_n\\)\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\\(n\\,\\,\\to\\,\\,\\infty\\)\uc77c \ub54c \\(x_n \\,\\,\\to\\,\\,L\\)\uc774\ub2e4.<\/li>\n<li>\\(x_n \\,\\,\\to\\,\\,L .\\)<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{x_n \\right\\}\\)\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{x_n \\right\\}\\)\uc758 \uadf9\ud55c\uc740 \\(L\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>(4)\uc640 \uac19\uc774 \ub4f1\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \uac83\uc740 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc774 \uc720\uc77c\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uc720\uc77c\ud558\ub2e4. \uc989 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\uace0 \\(L_1\\)\uacfc \\(L_2\\)\uac00 \\(\\left\\{x_n \\right\\}\\)\uc758 \uadf9\ud55c\uc774\uba74 \\(L_1 = L_2\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\frac{\\epsilon}{2}\\)\ub3c4 \uc591\uc218\uc774\ubbc0\ub85c \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert x_n &#8211; L_1 \\right\\rvert < \\frac{\\epsilon}{2}\\tag{5}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba70, \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_2\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert x_n &#8211; L_2 \\right\\rvert < \\frac{\\epsilon}{2}\\tag{6}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(N = \\max \\left\\{ N_1 ,\\, N_2 \\right\\}\\)\ub77c\uace0 \ud558\uace0 \\(n > N\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc744 \ud0dd\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\left\\lvert L_1 &#8211; L_2 \\right\\rvert<br \/>\n&#038;= \\left\\lvert ( L_1 &#8211; x_n ) + (x_n &#8211; L_2 ) \\right\\rvert \\\\[8pt]<br \/>\n&#038;\\le \\left\\lvert L_1 &#8211; x_n \\right\\rvert + \\left\\lvert x_n &#8211; L_2 \\right\\rvert \\\\[8pt]<br \/>\n&#038;< \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\\\\[8pt]\n\\end{align}\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\epsilon\\)\uc774 \uc784\uc758\uc758 \uc591\uc218\uc774\ubbc0\ub85c \\(L_1 = L_2\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub9cc\uc57d [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\le u\\)]\uc778 \uc2e4\uc218 \\(u\\)\uac00 \uc874\uc7ac\ud558\uba74 \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uba70, \ub9cc\uc57d [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\ge \\ell\\)]\uc778 \uc2e4\uc218 \\(\\ell\\)\uc774 \uc874\uc7ac\ud558\uba74 \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub610\ud55c \\(\\left\\{x_n \\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc77c \ub54c \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc720\uacc4<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uc720\uacc4\uc131)<\/span><\/p>\n<p>\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \uc720\uacc4\uc774\ub2e4. \uc989 \\(\\left\\{x_n \\right\\}\\)\uc774 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\uba74 \uc2e4\uc218 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert x_n \\right\\rvert \\le B\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc758 \uadf9\ud55c\uc774 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon = 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\lvert x_n &#8211; L \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[B = \\max \\left\\{ \\left\\lvert x_1 \\right\\rvert ,\\, \\left\\lvert x_2 \\right\\rvert ,\\, \\cdots ,\\, \\left\\lvert x_N \\right\\rvert ,\\, \\lvert L \\rvert + \\epsilon \\right\\}\\]\n\uc774\ub77c\uace0 \ud558\uba74 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert x_n \\right\\rvert \\le B\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc758 \uc77c\ubd80 \ud56d\uc744 \uc21c\uc11c\ub300\ub85c \ub098\uc5f4\ud558\uc5ec<br \/>\n\\[x_2 ,\\, x_4 ,\\, x_6 ,\\, x_8 ,\\, \\cdots \\tag{7}\\]<br \/>\n\uacfc \uac19\uc774 \ub610 \ud558\ub098\uc758 \uc218\uc5f4\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub294\ub370, \uc774\ub7ec\ud55c \uc218\uc5f4\uc744 \ubd80\ubd84\uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(n_k = 2k\\)\ub77c\uace0 \ud558\uba74 (7)\uc740 \\(\\left\\{ x_{n_k} \\right\\}\\)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc989 \\(\\left\\{ n_k \\right\\}\\)\uac00 \uc99d\uac00\ud558\ub294 \uc790\uc5f0\uc218\uc5f4\uc77c \ub54c \ud569\uc131\ud568\uc218 \\(\\left\\{ x_{n_k} \\right\\}\\)\ub97c \\(\\left\\{ x_n \\right\\}\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\ubb3c\ub860 \\(\\left\\{ x_n \\right\\}\\)\uc740 \uc790\uae30 \uc790\uc2e0\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4.]<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \ubd80\ubd84\uc218\uc5f4)<\/span><\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc774\uace0 \uadf8 \uadf9\ud55c\uac12\uc774 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc758 \uc784\uc758\uc758 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ x_{n_k} \\right\\}\\)\ub294 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\lvert x_n &#8211; L \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\left\\{ n_k \\right\\}\\)\uac00 \uc99d\uac00\ud558\ub294 \uc790\uc5f0\uc218\uc5f4\uc774\uace0, \\(\\mathbb{N}\\)\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \\(n_N \\ge N_1\\)\uc778 \ucca8\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(k > N\\)\uc774\uba74 \\(n_k > n_N \\ge N_1\\)\uc774\ubbc0\ub85c \\(\\left\\lvert x_{n_k} &#8211; L \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ x_{n_k}\\right\\}\\)\ub294 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"algebraicproperty\"><\/a><\/p>\n<h3>\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8<\/h3>\n<p>\uadf9\ud55c\uc758 \uc815\uc758\ub294 \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc9c1\uc811\uc801\uc73c\ub85c \uc81c\uacf5\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc774\ubbf8 \uadf9\ud55c\uac12\uc744 \uc54c\uace0 \uc788\ub294 \uc218\uc5f4\uc744 \ub300\uc218\uc801\uc73c\ub85c \ubcc0\ud615\ud558\uac70\ub098 \uacb0\ud569\ud558\uc5ec \uc0c8\ub85c\uc6b4 \uc218\uc5f4\uc744 \ub9cc\ub4e4\uc5c8\uc744 \ub54c \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uadf9\ud55c\uac12\uc744 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8)<\/span><\/p>\n<p>\ub9cc\uc57d \\(L,\\) \\(M,\\) \\(k\\)\uac00 \uc2e4\uc218\uc774\uace0<br \/>\n\\[\\lim_{n\\to\\infty} x_n = L , \\quad \\lim_{n\\to\\infty}y_n =M\\]<br \/>\n\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\left\\{ x_n + y_n \\right\\}\\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim _ { n\\to\\infty} (x_n + y_n) = L+M.\\]<\/li>\n<li>\\(\\left\\{ x_n &#8211; y_n \\right\\}\\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim _ {n\\to\\infty} (x_n &#8211; y_n) = L-M .\\]<\/li>\n<li>\\(\\left\\{ kx_n \\right\\} \\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim _ { n\\to\\infty} (k x_n) = kL .\\]<\/li>\n<li>\\(\\left\\{ x_n y_n \\right\\}\\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim _ { n\\to\\infty} (x_n y_n) = LM .\\]<\/li>\n<li>\\(M \\ne 0\\)\uc774\uba74 \\(\\left\\{ x_n \/ y_n \\right\\}\\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{x_n}{y_n} = \\frac{L}{M} .\\]<\/li>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p><span class=\"proof\">[1]\uc758 \uc99d\uba85.<\/span> \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\frac{\\epsilon}{2} > 0\\)\uc774\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\lvert x_n &#8211; L \\right\\rvert < \\frac{\\epsilon}{2}\\)\uc774 \uc131\ub9bd\ud558\uba70, \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_2\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\lvert y_n &#8211; M \\right\\rvert < \\frac{\\epsilon}{2}\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(N = \\,\\)\\(\\max\\left\\{ N_1 ,\\, N_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\begin{align}<br \/>\n\\left\\lvert \\left(x_n + y_n\\right) &#8211; (L+M) \\right\\rvert<br \/>\n&#038;= \\left\\lvert \\left(x_n -L\\right) + \\left( y_n -M\\right) \\right\\rvert \\\\[8pt]<br \/>\n&#038;\\le \\left\\lvert x_n -L \\right\\rvert + \\left\\lvert y_n -M \\right\\rvert \\\\[6pt]<br \/>\n&#038;< \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(( x_n + y_n ) \\,\\to\\, (L+M)\\)\uc774\ub2e4.<\/p>\n<p><span class=\"proof\">[4]\uc758 \uc99d\uba85.<\/span> \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(1\\)\uc774 \uc591\uc218\uc774\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4 \\[\\left\\lvert x_n &#8211; L \\right\\rvert < 1\\tag{8}\\]\uc774 \uc131\ub9bd\ud558\uba70, \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_2\\)\uc77c \ub54c\ub9c8\ub2e4 \\[\\left\\lvert y_n &#8211; M \\right\\rvert < 1\\tag{9}\\]\uc774 \uc131\ub9bd\ud55c\ub2e4. (8)\uacfc (9)\ub97c \uac01\uac01 \ubcc0\ud615\ud558\uba74\n\\[\\left\\lvert x_n \\right\\rvert < \\lvert L \\rvert +1 ,\\quad \\left\\lvert y_n \\right\\rvert < \\lvert M \\rvert +1\\tag{10}\\]\n\uc744 \uc5bb\ub294\ub2e4. \ud55c\ud3b8 \\(\\epsilon \/ (2(\\lvert L \\rvert +1 ))\\)\uacfc \\(\\epsilon \/ (2(\\lvert M \\rvert +1 ))\\)\uc740 \ubaa8\ub450 \uc591\uc218\uc774\ubbc0\ub85c \uc790\uc5f0\uc218 \\(N_3\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_3\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert x_n &#8211; L \\right\\rvert < \\frac{\\epsilon}{2(\\lvert M \\rvert + 1)}\\tag{11}\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \uc790\uc5f0\uc218 \\(N_4\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_4\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert y_n &#8211; M \\right\\rvert < \\frac{\\epsilon}{2(\\lvert L \\rvert + 1)}\\tag{12}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(N = \\max\\left\\{ N_1 ,\\, N_2 ,\\, N_3 ,\\, N_4 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n > N\\)\uc77c \ub54c (10), (11), (12)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\left\\lvert x_n y_n &#8211; LM \\right\\rvert<br \/>\n&#038;= \\left\\lvert x_n y_n &#8211; Ly_n + Ly_n &#8211; LM \\right\\rvert \\\\[8pt]<br \/>\n&#038;= \\left\\lvert (x_n &#8211; L)y_n + L(y_n &#8211; M) \\right\\rvert \\\\[8pt]<br \/>\n&#038;\\le \\left\\lvert x_n &#8211; L \\right\\rvert \\left\\lvert y_n \\right\\rvert + \\lvert L \\rvert \\left\\lvert y_n &#8211; M \\right\\rvert \\\\[6pt]<br \/>\n&#038;< \\frac{\\epsilon}{2(\\lvert M \\rvert + 1)} \\cdot (\\lvert M \\rvert + 1) + (\\lvert L \\rvert + 1) \\cdot \\frac{\\epsilon}{2(\\lvert L \\rvert + 1)}\\\\[6pt]\n&#038;= \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><span class=\"proof\">[3]\uc758 \uc99d\uba85.<\/span> \\(y_n = k\\)\ub77c\uace0 \ud558\uba74<br \/>\n\\[\\lim_{n\\to\\infty} y_n = k\\]<br \/>\n\uc774\ubbc0\ub85c [4]\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{n\\to\\infty} \\left( kx_n \\right) = \\lim_{n\\to\\infty} \\left( y_n x_n \\right) = kL\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><span class=\"proof\">[2]\uc758 \uc99d\uba85.<\/span> \uba3c\uc800 [3]\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{n\\to\\infty} \\left( -y_n \\right) = \\lim_{n\\to\\infty} \\left( (-1)y_n \\right)<br \/>\n= (-1)M = -M\\]<br \/>\n\uc774\ubbc0\ub85c [1]\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lim_{n\\to\\infty} \\left( x_n &#8211; y_n \\right) = \\lim_{n\\to\\infty} \\left( x_n + \\left( -y_n \\right) \\right) = L + (-M) = L-M\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p><span class=\"proof\">[5]\uc758 \uc99d\uba85.<\/span> <\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\frac{\\lvert M\\rvert}{2}\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert y_n &#8211; M \\right\\rvert < \\frac{\\lvert M \\rvert}{2}\\tag{13}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \\(\\frac{\\lvert M \\rvert ^2 \\epsilon}{2}\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_2\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\left\\lvert y_n &#8211; M \\right\\rvert < \\frac{\\lvert M \\rvert ^2}{2} \\epsilon\\tag{14}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(N = \\max\\left\\{ N_1 ,\\,N_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. (13)\uc744 \ubcc0\ud615\ud558\uba74\n\\[\\left\\lvert y_n \\right\\rvert > \\frac{\\lvert M\\rvert}{2}\\]<br \/>\n\uc989<br \/>\n\\[\\frac{1}{\\left\\lvert y_n \\right\\rvert} < \\frac{2}{\\lvert M \\rvert}\\tag{15}\\]\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (15), (14)\uc5d0 \uc758\ud558\uc5ec\n\\[\\begin{align}\n\\left\\lvert \\frac{1}{y_n} - \\frac{1}{M} \\right\\rvert &#038;= \\frac{\\left\\lvert M - y_n \\right\\rvert}{\\left\\lvert y_n \\right\\rvert\\,\\lvert M\\rvert} \\\\[6pt]\n&#038;< \\frac{2}{\\lvert M \\rvert^2} \\left\\lvert M-y_n \\right\\rvert \\\\[6pt]\n&#038;< \\frac{2}{\\lvert M \\rvert^2} \\cdot \\frac{\\lvert M \\rvert ^2}{2} \\epsilon = \\epsilon\n\\end{align}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989\n\\[\\lim_{n\\to\\infty}\\frac{1}{y_n} = \\frac{1}{M}\\]\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uacfc [2]\ub97c \uc774\uc6a9\ud558\uba74\n\\[\\lim_{n\\to\\infty}\\frac{x_n}{y_n} = \\lim_{n\\to\\infty}\\left( x_n \\cdot \\frac{1}{y_n}\\right) = L \\cdot \\frac{1}{M} = \\frac{L}{M}\\]\n\uc744 \uc5bb\ub294\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"analyticproperty\"><\/a><\/p>\n<h3>\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8<\/h3>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ubd80\ub4f1\ud638\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ub2e4\uc74c \uc815\ub9ac\ub294 \ud55c \uac12\uc5d0 \uc218\ub834\ud558\ub294 \ub450 \uc218\uc5f4 \uc0ac\uc774\uc5d0 \ub07c\uc5b4 \uc788\ub294 \uc218\uc5f4\uc740 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud568\uc744 \uc124\uba85\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 5. (\uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac)<\/span><\/p>\n<p>\\(\\left\\{ \\alpha_n \\right\\},\\) \\(\\left\\{ \\beta_n \\right\\},\\) \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\uace0, \uc790\uc5f0\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > M\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\alpha_n \\le x_n \\le \\beta_n\\]<br \/>\n\uc774 \uc131\ub9bd\ud558\uba70<br \/>\n\\[\\lim_{n\\to\\infty} \\alpha_n = \\lim_{n\\to\\infty} \\beta_n = L\\]<br \/>\n\uc774\uba74<br \/>\n\\[\\lim_{n\\to\\infty} x_n = L\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert \\alpha_n &#8211; L \\right\\rvert < \\epsilon\\]\n\uc989\n\\[L - \\epsilon < \\alpha_n < L+ \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\uba70, \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_2\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert \\beta_n &#8211; L \\right\\rvert < \\epsilon\\]\n\uc989\n\\[L - \\epsilon < \\beta_n < L+ \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(N = \\max\\left\\{M ,\\, N_1 ,\\, N_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n > N\\)\uc77c \ub54c<br \/>\n\\[L &#8211; \\epsilon < \\alpha_n \\le x_n \\le \\beta_n < L+\\epsilon\\]\n\uc989\n\\[\\left\\lvert x_n - L \\right\\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \\(x_n \\,\\to\\, L\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\le x_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc744 <span class=\"defined\">\uc99d\uac00\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \ub9cc\uc57d \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\ge x_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc744 <span class=\"defined\">\uac10\uc18c\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n < x_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc744 <span class=\"defined\">\uc21c\uc99d\uac00\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \ub9cc\uc57d \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n > x_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc744 <span class=\"defined\">\uc21c\uac10\uc18c\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc99d\uac00\uc218\uc5f4\uacfc \uac10\uc18c\uc218\uc5f4\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\ub2e8\uc870\uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 6. (\uc218\uc5f4\uc758 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc720\uacc4\uc778 \ub2e8\uc870\uc218\uc5f4\uc774\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc99d\uac00\uc218\uc5f4\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<br \/>\n\\[E = \\left\\{ x_n \\,\\vert\\, n\\in\\mathbb{N}\\right\\}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74 \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc0c1\ud55c\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(L &#8211; \\epsilon < x_N \\le L\\)\uc778 \\(x_N \\in E\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc99d\uac00\uc218\uc5f4\uc774\ubbc0\ub85c \\(n > N\\)\uc774\uba74<br \/>\n\\[L &#8211; \\epsilon < x_n \\le L < L + \\epsilon\\]\n\uc989\n\\[\\left\\lvert x_n - L \\right\\rvert < \\epsilon\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ x_n \\right\\}\\)\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc99d\uac00\ud558\uba74\uc11c \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc744 \\(x_n \\,\\nearrow\\,L\\)\ub85c \ub098\ud0c0\ub0b4\uba70, \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uac10\uc18c\ud558\uba74\uc11c \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc744 \\(x_n \\,\\searrow\\,L\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc720\uacc4\uc778 \uc218\uc5f4\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc815\ub9ac 2\uc5d0 \uc758\ud558\uba74 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \uc720\uacc4\uc774\ub2e4. \ud558\uc9c0\ub9cc \uadf8 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300 \\(x_n = (-1)^n\\)\uc774\ub77c\uace0 \ud558\uba74 \\(\\left\\{ x_n \\right\\}\\)\uc740 \uc720\uacc4\uc774\uc9c0\ub9cc \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc720\uacc4\uc778 \uc218\uc5f4\uc740 \ud56d\uc0c1 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 7. (\uc218\uc5f4\uc5d0 \ub300\ud55c \ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac)<\/span><\/p>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\uba74 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ x_{n_k} \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\left\\{ x_{n_k} \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc758 <span class=\"defined\">\uc9d1\uc801\uc810<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\ubbc0\ub85c \\(M > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in [-M ,\\, M]\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\(I_0 = [ -M ,\\, M]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(I_0\\)\uc744 \uc798\ub77c \uae38\uc774\uac00 \uac19\uc740 \ub450 \uac1c\uc758 \ub2eb\ud78c \uad6c\uac04 \\([-M ,\\, 0]\\)\uacfc \\([0,\\,M]\\)\uc744 \ub9cc\ub4e4\uc5c8\uc744 \ub54c \ub450 \uad6c\uac04 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c0\uac8c \ub418\ub294\ub370, \uadf8 \uad6c\uac04\uc744 \\(I_1 = \\left[\\alpha _1 ,\\, \\beta_1 \\right]\\)\uc774\ub77c\uace0 \ud558\uc790. [\ub9cc\uc57d \\([-M ,\\, 0]\\)\uacfc \\([0,\\,M]\\) \ubaa8\ub450 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c4\ub2e4\uba74, \ub458 \uc911 \uc5b4\ub290 \uac83\uc744 \\(I_1\\)\ub85c \ub450\ub4e0 \uc0c1\uad00 \uc5c6\ub2e4.] \ub2e4\uc2dc \\(I_1\\)\uc744 \uc798\ub77c \uae38\uc774\uac00 \uac19\uc740 \ub450 \uac1c\uc758 \ub2eb\ud78c \uad6c\uac04<br \/>\n\\[\\left[\\alpha_1 ,\\, \\frac{\\alpha_1 + \\beta_1}{2}\\right],\\quad \\left[\\frac{\\alpha_1 + \\beta_1}{2},\\, \\beta_1 \\right]\\]<br \/>\n\uc744 \ub9cc\ub4e4\uc5c8\uc744 \ub54c \ub450 \uad6c\uac04 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c0\ub294\ub370, \uadf8 \uad6c\uac04\uc744 \\(I_2 = \\left[\\alpha_2 ,\\, \\beta_2 \\right]\\)\ub77c\uace0 \ud558\uc790. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec(\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec) \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud78c \uad6c\uac04 \\(I_k\\)\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uba85\ubc31\ud788 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha_k \\le \\alpha_{k+1} \\le \\beta_{k+1} \\le \\beta_k\\)\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ \\alpha_k \\right\\}\\)\uc640 \\(\\left\\{ \\beta_k \\right\\}\\)\ub294 \uac01\uac01 \uc218\ub834\ud55c\ub2e4. \ubfd0\ub9cc \uc544\ub2c8\ub77c \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\beta_k &#8211; \\alpha_k = \\frac{2M}{2^k}\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\left( \\beta_k &#8211; \\alpha_k \\right) \\to 0\\)\uc774\ub2e4. \uc989 \\(\\left\\{ \\alpha_k \\right\\}\\)\uc640 \\(\\left\\{ \\beta_k \\right\\}\\)\ub294 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(I_1\\)\uc5d0 \uc18d\ud558\ub294 \\(\\left\\{x_n \\right\\}\\)\uc758 \ud56d\uc744 \ud558\ub098 \ud0dd\ud558\uc5ec \\(x_{n_1}\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc73c\ub85c \\(\\left\\{x_n \\right\\}\\)\uc758 \ud56d \uc911\uc5d0\uc11c \\(I_2\\)\uc5d0 \uc18d\ud558\uba74\uc11c \ucca8\uc218\uac00 \\(n_1\\)\ubcf4\ub2e4 \ud070 \ud56d\uc744 \ud558\ub098 \ud0dd\ud558\uc5ec \\(x_{n_2}\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc2dc \\(I_3\\)\uc5d0 \uc18d\ud558\uba74\uc11c \ucca8\uc218\uac00 \\(n_2\\)\ubcf4\ub2e4 \ud070 \ud56d\uc744 \ud558\ub098 \ud0dd\ud558\uc5ec \\(x_{n_3}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ x_{n_k}\\right\\}\\)\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\n<p>\\(\\left\\{ x_{n_k}\\right\\}\\)\uc758 \uad6c\uc131 \uacfc\uc815\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha_k \\le x_k \\le \\beta_k\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\left\\{ \\alpha_k \\right\\}\\)\uc640 \\(\\left\\{ \\beta_k \\right\\}\\)\uac00 \ubaa8\ub450 \\(L\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ x_{n_k}\\right\\}\\)\ub3c4 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub05d\uc73c\ub85c \ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub41c \uc218\uc5f4\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 8. (\ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c\uc758 \uc218\uc5f4\uc758 \uadf9\ud55c)<\/span><\/p>\n<p>\\(I = [a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c \uad6c\uac04\uc774\uace0 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(I\\)\uc5d0 \uc18d\ud558\uba70 \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(L \\in I\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(L\\notin I\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(L < a\\)\uc774\uac70\ub098 \\(b < L\\)\uc774\ub2e4.\n\\[\\epsilon = \\min\\left\\{ \\lvert a-L \\rvert ,\\, \\lvert L-b \\rvert \\right\\}\\]\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\epsilon > 0\\)\uc774\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\[\\left\\lvert x_n &#8211; L \\right\\rvert < \\epsilon\\tag{16}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. (16)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud56d \\(x_n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n < a\\)\uc774\uac70\ub098 \\(b < x_n\\)\uc774\ubbc0\ub85c, \\(x_n\\)\uc740 \\(I\\) \ubc14\uae65\uc5d0 \ub193\uc778\ub2e4. \uc774\uac83\uc740 \\(\\left\\{ x_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(I\\)\uc5d0 \uc18d\ud55c\ub2e4\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(L \\in I\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"divergent\"><\/a><\/p>\n<h3>\ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4<\/h3>\n<p>\uc55e\uc5d0\uc11c \uc815\uc758\ud55c \ubc14\uc640 \uac19\uc774 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \uc5b4\ub5a0\ud55c \uac12\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u2018\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\ub3c4 \uba87 \uac00\uc9c0\ub85c \ubd84\ub958\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4)<\/span><\/p>\n<p>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(x_n > B\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294 \\(x_n \\,\\,\\to\\,\\, \\infty\\) \ub610\ub294\\[\\lim_{n\\to\\infty} x_n = \\infty\\]\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(B > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(x_n < -B\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294 \\(x_n \\,\\,\\to\\,\\, -\\infty\\) \ub610\ub294\\[\\lim_{n\\to\\infty} x_n = -\\infty\\]\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(\\left\\{ x_n \\right\\}\\)\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\uc73c\uba74 \u2018\\(\\left\\{ x_n \\right\\}\\)\uc740 <span class=\"defined\">\uc9c4\ub3d9<\/span>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uba74 \uc591\uc758 \ubb34\ud55c\ub300\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba70, \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989 \u2018\uc218\ub834\u2019, \u2018\uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\u2019, \u2018\uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\u2019, \u2018\uc9c4\ub3d9\u2019\uc740 \uc0c1\ud638 \ubca0\ud0c0\uc801\uc778 \uacbd\uc6b0\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc740 \ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4\uc758 \uc131\uc9c8 \uc911 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uac83\ub4e4\uc774\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li><span class=\"definition\">\ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4\uc758 \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac<\/span>:&nbsp; \\(\\left\\{ x_n \\right\\}\\)\uacfc \\(\\left\\{ y_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\uace0, \uc790\uc5f0\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > M\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\le y_n\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x_n \\,\\to\\, \\infty\\)\uc774\uba74 \\(y_n \\,\\to\\, \\infty\\)\uc774\ub2e4. \ub9cc\uc57d \\(y_n \\,\\to\\, -\\infty\\)\uc774\uba74 \\(x_n \\,\\to\\, -\\infty\\)\uc774\ub2e4.<\/li>\n<li>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\left\\{ -x_n \\right\\}\\)\uc774 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uac83\uc774\ub2e4.<\/li>\n<li>\\(\\left\\{ x_n \\right\\}\\)\uacfc \\(\\left\\{ y_n \\right\\}\\)\uc774 \ubaa8\ub450 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4\uc774\uba74 \\(\\left\\{ x_n + y_n \\right\\}\\)\uacfc \\(\\left\\{ x_n y_n \\right\\}\\)\ub3c4 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<li>\\(\\left\\{ x_n \\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0 \\(\\left\\{ y_n \\right\\}\\)\uc774 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \\(\\left\\{ x_n + y_n \\right\\}\\)\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<p><!--\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p> ... \n\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0 \uadf9\ud55c\uacfc \uad00\ub828\ub41c \uae30\ubcf8\uc801\uc778 \uc131\uc9c8\uc744 \uc99d\uba85\ud55c\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758 \uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8 \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8 \ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 \uc9d1\ud569\uacfc \uc2e4\uc218 (\uad00\ub828 \uae00) \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758 \uc801\ub2f9\ud55c \uc815\uc218 \\(n_0\\)\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\uc5ed\uc744 \\(\\left\\{ n \\in \\mathbb{Z} \\,\\vert\\, n \\ge n_0 \\right\\}\\) \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ud568\uc218\ub97c \uc810\uc5f4(sequence)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc778 \uc810\uc5f4\uc744 \uc2e4\uc218\uc5f4(real sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. [\ubbf8\uc801\ubd84\ud559\uc744 \uacf5\ubd80\ud558\ub294&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[135,129,140,141,127,133,142,131,125,126,132,138,134,130,137,124,128,123,139,136],"class_list":["post-2837","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-bolzano-weierstrass","tag-boundedness","tag-convergence","tag-divergence","tag-limit","tag-monotone-convergence","tag-oscillation","tag-sandwich-theorem","tag-sequence","tag-126","tag-132","tag-138","tag-134","tag-130","tag-137","tag-124","tag-128","tag-123","tag-139","tag-136"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2837","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=2837"}],"version-history":[{"count":75,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2837\/revisions"}],"predecessor-version":[{"id":4301,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2837\/revisions\/4301"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2837"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2837"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2837"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}