{"id":6675,"date":"2021-07-20T23:49:52","date_gmt":"2021-07-20T14:49:52","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6675"},"modified":"2021-07-22T21:11:52","modified_gmt":"2021-07-22T12:11:52","slug":"bounded-and-monotone-sequences","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/bounded-and-monotone-sequences\/","title":{"rendered":"\uc720\uacc4\uc218\uc5f4\uacfc \ub2e8\uc870\uc218\uc5f4"},"content":{"rendered":"<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 5\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc720\uacc4\uc218\uc5f4<\/h2>\n<p>\ub2eb\ud78c\uad6c\uac04 \\(I = [-3,\\,4]\\)\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. \uba85\ubc31\ud788 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x \\rvert \\le 4\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub7ec\ud55c \ub9e5\ub77d\uc5d0\uc11c \u201c\\(I\\)\ub294 <span class=\"defined\">\uc720\uacc4<\/span>(bounded)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\uc774\ubc88\uc5d0\ub294 \uc9d1\ud569 \\(J = \\left\\{ x \\,\\vert\\, x \\ge 3 \\right\\}\\)\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. [\uc784\uc758\uc758 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x \\rvert \\le B\\)]\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(B\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774\ub54c \u201c\\(J\\)\ub294 <span class=\"defined\">\uc720\uacc4\uac00 \uc544\ub2c8\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ube44\ub85d \\(J\\)\uac00 \uc720\uacc4\uc778 \uc9d1\ud569\uc740 \uc544\ub2c8\uc9c0\ub9cc, \ub9cc\uc57d \\(m\\)\uc774 \\(3\\) \uc774\ud558\uc778 \uc2e4\uc218\ub77c\uba74 [\uc784\uc758\uc758 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(x \\ge m\\)]\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub7ec\ud55c \ub9e5\ub77d\uc5d0\uc11c \u201c\\(J\\)\ub294 \\(m\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>(bounded below)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(K\\)\uac00 \ubaa8\ub4e0 \uc74c\uc758 \uc2e4\uc218\uc758 \uc9d1\ud569\uc774\uace0 \\(M\\)\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc2e4\uc218\ub77c\uba74 [\uc784\uc758\uc758 \\(x\\in K\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\le M\\)]\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub7ec\ud55c \ub9e5\ub77d\uc5d0\uc11c \u201c\\(K\\)\ub294 \\(M\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>(bounded above)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.5.1.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\uc9d1\ud569 \\(A=\\left\\{ x\\in\\mathbb{R} \\,\\vert\\, -4 \\le x < 5 \\right\\}\\)\ub294 \uc720\uacc4\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{R}\\)\uc758 \uc784\uc758\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569\uc740 \uc720\uacc4\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{N}\\)\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\uc9c0\ub9cc \uc704\ub85c\ub294 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\mathbb{N}\\)\uc740 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(\\mathbb{Q}\\)\ub294 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba70 \uc544\ub798\ub85c\ub3c4 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc218\uc5f4\uc758 \uc720\uacc4\uc131\ub3c4 \uc9d1\ud569\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\uc758\ub41c\ub2e4. \uc989 \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ub9cc\uc57d \uc2e4\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uc5ec [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\ge m\\)]\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \u201c\\(\\left\\{a_n\\right\\}\\)\uc740 \\(m\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>(bounded below)\uc774\ub2e4\u201d \ub610\ub294 \uac04\ub2e8\ud788 \u201c\\(\\left\\{a_n\\right\\}\\)\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le M\\)]\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \u201c\\(\\left\\{a_n\\right\\}\\)\uc740 \\(M\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>(bounded above)\uc774\ub2e4\u201d \ub610\ub294 \uac04\ub2e8\ud788 \u201c\\(\\left\\{a_n\\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\\(\\left\\{a_n\\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \u201c\\(\\left\\{a_n\\right\\}\\)\uc740 <span class=\"defined\">\uc720\uacc4<\/span>(bounded)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0 \\(\\left\\{a_n\\right\\}\\)\uc744 <span class=\"defined\">\uc720\uacc4\uc218\uc5f4<\/span>(bounded sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.5.2.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\uc218\uc5f4 \\(\\left\\{ n \\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uc9c0\ub9cc \uc544\ub798\ub85c\ub294 \uc720\uacc4\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{\\frac{n+1}{n}\\right\\}\\)\uc740 \uc720\uacc4\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\[0\\le\\frac{n+1}{n}\\le 1\\]\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{ (-1)^n \\right\\}\\)\uc740 \uc720\uacc4\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc784\uc758\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\[-1 \\le (-1)^n \\le 1\\]\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{ (-2)^n\\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uace0 \uc544\ub798\ub85c\ub3c4 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc0c1\uc218\uc218\uc5f4\uc740 \uc720\uacc4\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\\(\\left\\{ a_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(n\\)\uc774 \ucee4\uc9d0\uc5d0 \ub530\ub77c \\(a_n\\)\uc740 \\(L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc774 \ub2e4\uac00\uac00\ubbc0\ub85c, \uc720\ud55c \uac1c\ub97c \uc81c\uc678\ud55c \\(n\\)\uc5d0 \ub300\ud574\uc11c<br \/>\n\\[L-1 \\le a_n \\le L+1\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec [\\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(L &#8211; 1 \\le a_n \\le L+1 \\)]\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(N+1\\)\uac1c\uc758 \uc218<br \/>\n\\[\\left\\lvert a_1 \\right\\rvert ,\\,\\, \\left\\lvert a_2 \\right\\rvert ,\\,\\, \\left\\lvert a_3 \\right\\rvert ,\\,\\, \\cdots ,\\,\\, \\left\\lvert a_N \\right\\rvert ,\\,\\, \\lvert L \\rvert +1\\]<br \/>\n\uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc744 \ud0dd\ud558\uc5ec \\(M\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\left\\lvert a_n \\right\\rvert \\le M\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.5.1. (\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \uc720\uacc4\uc131)<\/span><\/p>\n<p>\ub9cc\uc57d \uc2e4\uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74, \uadf8 \uc218\uc5f4\uc740 \uc720\uacc4\uc218\uc5f4\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><span class=\"remark\">\uc8fc\uc758.<\/span> \uc720\uacc4\uc778 \uc218\uc5f4\uc774 \ubaa8\ub450 \uc218\ub834\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ucee8\ub300  \\(a_n = (-1)^n\\)\uc774\ub77c\uace0 \ud558\uba74 \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc740 \uc720\uacc4\uc774\uc9c0\ub9cc \uc218\ub834\ud558\uc9c0 \uc54a\uace0 \uc9c4\ub3d9\ud55c\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ub2e8\uc870\uc218\uc5f4<\/h2>\n<p>\uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774 \ud56d\uc0c1 \uc218\ub834\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774 \u2018\ub2e8\uc870\u2019\uc218\uc5f4\uc774\ub77c\uba74 \uadf8 \uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4. \uc774\uac83\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud558\uc5ec \uba87 \uac00\uc9c0 \uac1c\ub150\uc744 \ub3c4\uc785\ud55c\ub2e4.<\/p>\n<p>\\(\\left\\{ a_n \\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \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, \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 <span class=\"defined\">\ub2e8\uc870\uc99d\uac00\ud55c\ub2e4<\/span>(increases monotonically)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\ge a_{n+1}\\)\uc774 \uc131\ub9bd\ud558\uba74, \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 <span class=\"defined\">\ub2e8\uc870\uac10\uc18c\ud55c\ub2e4<\/span>(decreases monotonically)\u201d\ub77c\uace0 \ub9d0\ud55c\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>(monotonic sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.5.3.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\uc218\uc5f4 \\(\\left\\{ n \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{ \\frac{n+1}{n} \\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{ (-1)^n \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uc9c0\ub3c4 \uc54a\uace0 \ub2e8\uc870\uac10\uc18c\ud558\uc9c0\ub3c4 \uc54a\ub294\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc0c1\uc218\uc218\uc5f4\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\uc778 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le M\\)]\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218 \\(M\\) \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac83\uc744 \ud0dd\ud558\uc790. (\uadf8\ub7ec\ud55c \\(M\\)\uc744 \\(\\left\\{ a_n \\right\\}\\)\uc758 <span class=\"defined\">\ucd5c\uc18c\uc0c1\uacc4<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.) \uadf8\ub7ec\uba74 \\(\\left\\{a_n \\right\\}\\)\uc740 \\(M\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(\\left\\{a_n \\right\\}\\)\uc774 \\(M\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 [\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le M\\)]\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub354 \uc791\uc740 \uc218 \\(M\\)\uc744 \ud0dd\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\uac19\uc740 \ubc29\ubc95\uc73c\ub85c, \ub9cc\uc57d \\(\\left\\{ b_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \uc544\ub798\ub85c \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\ub77c\uba74 \\(\\left\\{ b_n \\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. [\uc774 \uacbd\uc6b0 \\(\\left\\{ b_n \\right\\}\\)\uc740 \uc790\uc2e0\uc758 <span class=\"defined\">\ucd5c\ub300\ud558\uacc4<\/span>\uc5d0 \uc218\ub834\ud55c\ub2e4.]<\/p>\n<p>\uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.5.2. (\ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p>\ub2e8\uc870\uc774\uba74\uc11c \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub294 \uadf9\ud55c\uac12\uc744 \uc54c\uc9c0 \ubabb\ud558\ub294 \uc0c1\ud0dc\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834\uc744 \uc99d\uba85\ud560 \ub54c \uc720\uc6a9\ud558\uac8c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 1.5.4.<\/span><br \/>\n\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[a_1 = 1 \\quad \\text{and} \\quad a_{n+1} = \\sqrt{2+a_n}.\\]<br \/>\n\uc774\ub54c \\(\\left\\{a_n\\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uace0, \uadf8 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span> \\(a_1 = 1 \\le 2\\)\uc774\uace0<br \/>\n\\[a_n \\le 2 \\quad\\Rightarrow\\quad a_{n+1} = \\sqrt{2+a_n} \\le \\sqrt{2+2} = 2\\]<br \/>\n\uc774\ubbc0\ub85c, \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le 2\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \\(\\left\\{a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\uace0<br \/>\n\\[\\left( a_{n+1}\\right)^2 = 2+a_n \\ge a_n + a_n = 2a_n \\ge a_n \\cdot a_n = \\left(a_n \\right)^2\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\left\\{a_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{a_n\\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\\(\\left\\{a_n\\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[L = \\lim_{n\\rightarrow\\infty} a_n = \\lim_{n\\rightarrow\\infty}a_{n+1} = \\sqrt{2+\\lim_{n\\rightarrow\\infty} a_n} = \\sqrt{2+L}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(L \\ge 0\\)\uc774\ubbc0\ub85c \uc774 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \\(L=2\\)\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{a_n\\right\\}\\)\uc740 \\(2\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc218\uc5f4\uc774 \uadc0\ub0a9\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c, \uc218\uc5f4\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \ubc1d\ud788\uc9c0 \uc54a\uace0 \uadf9\ud55c\uac12\uc744 \uad6c\ud558\uba74 \uc798\ubabb\ub41c \uacb0\ub860\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc608\ub97c \ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.5.5.<\/span><br \/>\n\uc218\uc5f4 \\(\\left\\{ b_n \\right\\}\\)\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[b_1 = b_2 = 1 \\quad\\text{and}\\quad b_{n+2} = b_{n+1} + b_n .\\]<br \/>\n\ub9cc\uc57d \\(\\left\\{ b_n \\right\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0(\ubb3c\ub860 \uc798\ubabb\ub41c \uac00\uc815\uc774\ub2e4) \uadf8 \uadf9\ud55c\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uba74 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[L = \\lim_{n\\rightarrow\\infty}b_{n+2} = \\lim_{n\\rightarrow\\infty}\\left(b_{n+1} + b_n\\right) = \\lim _ {n\\rightarrow\\infty} b_{n+1} + \\lim_{n\\rightarrow\\infty} b_n = L+L = 2L\\]<br \/>\n\uc774 \ub4f1\uc2dd \\(L = 2L\\)\uc744 \ud480\uba74 \\(L=0\\)\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(b_n \\ge n-1\\)\uc774\ubbc0\ub85c, \\(\\left\\{ b_n \\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\uace0 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\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<\/div>\n\n\n\n\n########\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}<br \/>\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}<br \/>\n\\newcommand{\\proj}{{\\operatorname{proj}}}<br \/>\n\\newcommand{\\adj}{{\\operatorname{adj}}}<br \/>\n\\]\n<\/div>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/geometric-sequences\">\ub4f1\ube44\uc218\uc5f4\uc758 \uadf9\ud55c<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/limit-superior-and-limit-inferior\">\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 5\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc720\uacc4\uc218\uc5f4 \ub2eb\ud78c\uad6c\uac04 \\(I = [-3,\\,4]\\)\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. \uba85\ubc31\ud788 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x \\rvert \\le 4\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub7ec\ud55c \ub9e5\ub77d\uc5d0\uc11c \u201c\\(I\\)\ub294 \uc720\uacc4(bounded)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\ubc88\uc5d0\ub294 \uc9d1\ud569 \\(J = \\left\\{ x \\,\\vert\\, x \\ge 3 \\right\\}\\)\ub97c \uc0dd\uac01\ud574 \ubcf4\uc790. [\uc784\uc758\uc758 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert x \\rvert \\le B\\)]\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(B\\)\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774\ub54c \u201c\\(J\\)\ub294 \uc720\uacc4\uac00&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":105,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6675","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6675","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=6675"}],"version-history":[{"count":15,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6675\/revisions"}],"predecessor-version":[{"id":6892,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6675\/revisions\/6892"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}