{"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":"2026-09-27T17:14:17","modified_gmt":"2026-09-27T08:14:17","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>\\(E\\)\uac00 \uc2e4\uc218\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nx\\le M<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uba74 \\(E\\)\uac00 <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>(bounded above)\ub77c\uace0 \ub9d0\ud558\uace0 \\(M\\)\uc744 \\(E\\)\uc758 \uc0c1\uacc4\ub77c\uace0 \ubd80\ub978\ub2e4. \ubc18\ub300\ub85c \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nm\\le x<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uba74 \\(E\\)\uac00 <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>(bounded below)\ub77c\uace0 \ub9d0\ud558\uace0 \\(m\\)\uc744 \\(E\\)\uc758 \ud558\uacc4(lower bound)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(E\\)\uac00 \uc704\ub85c \uc720\uacc4\uc774\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \\(E\\)\uac00 <span class=\"defined\">\uc720\uacc4<\/span>(bounded)\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc740 \uc5b4\ub5a4 \uc591\uc218 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|x|\\le B<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uacfc \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4 \ub2eb\ud78c\uad6c\uac04<br \/>\n\\[<br \/>\nI=[-3,\\,4]<br \/>\n\\]<br \/>\n\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n|x|\\le4<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(I\\)\ub294 \uc720\uacc4\uc774\ub2e4. \ud55c\ud3b8<br \/>\n\\[<br \/>\nJ=\\left\\{x\\in\\mathbb{R}\\,\\middle|\\,x\\ge3\\right\\}<br \/>\n\\]<br \/>\n\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uc9c0\ub9cc \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \uc74c\uc758 \uc2e4\uc218\uc758 \uc9d1\ud569\uc740 \uc704\ub85c \uc720\uacc4\uc774\uc9c0\ub9cc \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\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<br \/>\n\\[<br \/>\nA=\\left\\{x\\in\\mathbb{R}\\,\\middle|\\,-4\\le x < 5\\right\\}\n\\]\n\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. \ub530\ub77c\uc11c \\(\\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 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\uc5b4\ub5a4 \uc2e4\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nm\\le a_n<br \/>\n\\]<br \/>\n\uc774\uba74 \\(\\left\\{a_n\\right\\}\\)\uc774 \\(m\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc544\ub798\ub85c \uc720\uacc4<\/span>\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\uc5b4\ub5a4 \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n\\le M<br \/>\n\\]<br \/>\n\uc774\uba74 \\(\\left\\{a_n\\right\\}\\)\uc774 \\(M\\)\uc5d0 \uc758\ud558\uc5ec <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>\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 \\(\\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<p>\ub530\ub77c\uc11c \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc218\uc5f4\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc5b4\ub5a4 \uc591\uc218 \\(B\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|a_n|\\le B<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\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 \uc720\uacc4\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4<br \/>\n\\[<br \/>\n\\left\\{\\frac{n+1}{n}\\right\\}<br \/>\n\\]<br \/>\n\uc740 \uc720\uacc4\uc774\ub2e4. \uc2e4\uc81c\ub85c \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n1\\le\\frac{n+1}{n}\\le2<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\uc218\uc5f4 \\(\\left\\{(-1)^n\\right\\}\\)\uc740 \uc720\uacc4\uc774\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[<br \/>\n-1\\le(-1)^n\\le1<br \/>\n\\]<br \/>\n\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<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>\uc2e4\uc218\uc5f4\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uc218\uc5f4\uc740 \uc720\uacc4\uc218\uc5f4\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(\\left\\{a_n\\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc218\ub834\uc758 \uc815\uc758\uc5d0\uc11c \\(\\epsilon=1\\)\ub85c \ub193\uc73c\uba74 \uc5b4\ub5a4 \ucca8\uc790 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N\\)\uc778 \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|a_n-L| < 1\n\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\n|a_n|\n\\le\n|a_n-L|+|L|\n<\n|L|+1.\n\\]\n<\/p>\n<p>\ud55c\ud3b8 \\(N\\)\ubcf4\ub2e4 \uc55e\uc5d0 \uc788\ub294 \ud56d\uc740 \uc720\ud55c \uac1c\ubfd0\uc774\ub2e4. \uadf8 \ud56d\ub4e4\uc758 \uc808\ub313\uac12\uacfc \\(|L|+1\\) \uac00\uc6b4\ub370 \uac00\uc7a5 \ud070 \uac12\uc744 \\(M\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n|a_n|\\le M.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\left\\{a_n\\right\\}\\)\uc740 \uc720\uacc4\uc218\uc5f4\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/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\ub97c \ub4e4\uc5b4<br \/>\n\\[<br \/>\na_n=(-1)^n<br \/>\n\\]<br \/>\n\uc774\uba74 \\(\\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. \uadf8\ub7ec\ub098 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774 \ub2e8\uc870\uc218\uc5f4\uc774\uba74 \ubc18\ub4dc\uc2dc \uc218\ub834\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>\ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n\\le a_{n+1}<br \/>\n\\]<br \/>\n\uc774\uba74 \\(\\left\\{a_n\\right\\}\\)\uc774 <span class=\"defined\">\ub2e8\uc870\uc99d\uac00\ud55c\ub2e4<\/span>(increases monotonically)\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n\\ge a_{n+1}<br \/>\n\\]<br \/>\n\uc774\uba74 \\(\\left\\{a_n\\right\\}\\)\uc774 <span class=\"defined\">\ub2e8\uc870\uac10\uc18c\ud55c\ub2e4<\/span>(decreases monotonically)\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>(monotone 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<br \/>\n\\[<br \/>\n\\left\\{\\frac{n+1}{n}\\right\\}<br \/>\n\\]<br \/>\n\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 \ub3d9\uc2dc\uc5d0 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc2e4\uc218\uc758 \uc9d1\ud569 \\(E\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \\(E\\)\uc758 \ubaa8\ub4e0 \ud558\uacc4 \uc911 \uac00\uc7a5 \ud070 \uac12\uc744 \\(E\\)\uc758 <span class=\"defined\">\ucd5c\ub300\ud558\uacc4<\/span>(greatest lower bound) \ub610\ub294 <span class=\"defined\">\ud558\ud55c<\/span>(infimum)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[<br \/>\n\\inf E<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc2e4\uc218\uacc4\uc758 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ucd5c\ub300\ud558\uacc4\ub3c4 \uc874\uc7ac\ud55c\ub2e4. \uc2e4\uc81c\ub85c<br \/>\n\\[<br \/>\n\\inf E<br \/>\n=<br \/>\n-\\sup\\{-x\\mid x\\in E\\}.<br \/>\n\\]\n<\/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<ol class=\"bracket\">\n<li>\ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uc740 \uadf8 \uc218\uc5f4\uc758 \ud56d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc758 \ucd5c\uc18c\uc0c1\uacc4\uc774\ub2e4.<\/li>\n<li>\ub2e8\uc870\uac10\uc18c\ud558\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uc740 \uadf8 \uc218\uc5f4\uc758 \ud56d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc758 \ucd5c\ub300\ud558\uacc4\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\ub530\ub77c\uc11c \ub2e8\uc870\uc774\uba74\uc11c \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>(1) \\(\\left\\{a_n\\right\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uc218\uc5f4\uc758 \ud56d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc744<br \/>\n\\[<br \/>\nE=\\{a_n\\}<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uace0<br \/>\n\\[<br \/>\nL=\\sup E<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc2e4\uc218\uacc4\uc758 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(L\\)\uc740 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(L-\\epsilon\\)\uc740 \\(E\\)\uc758 \uc0c1\uacc4\uac00 \ub420 \uc218 \uc5c6\ub2e4. \ub9cc\uc57d \uc0c1\uacc4\ub77c\uba74 \\(L\\)\ubcf4\ub2e4 \uc791\uc740 \uc0c1\uacc4\uac00 \uc874\uc7ac\ud558\uac8c \ub418\uc5b4 \\(L\\)\uc774 \ucd5c\uc18c\uc0c1\uacc4\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \ucca8\uc790 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\nL-\\epsilon < a_N.\n\\]\n<\/p>\n<p>\uc218\uc5f4\uc774 \ub2e8\uc870\uc99d\uac00\ud558\ubbc0\ub85c \\(n\\ge N\\)\uc774\uba74<br \/>\n\\[<br \/>\na_N\\le a_n.<br \/>\n\\]<br \/>\n\ub610\ud55c \\(L\\)\uc740 \\(E\\)\uc758 \uc0c1\uacc4\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\na_n\\le L.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(n\\ge N\\)\uc774\uba74<br \/>\n\\[<br \/>\nL-\\epsilon<br \/>\n<\na_N\n\\le\na_n\n\\le\nL,\n\\]\n\uc989\n\\[\n|a_n-L| < \\epsilon.\n\\]\n\uadf8\ub7ec\ubbc0\ub85c\n\\[\na_n\\rightarrow L.\n\\]\n<\/p>\n<p>(2) \\(\\left\\{a_n\\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \uc218\uc5f4\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[<br \/>\nb_n=-a_n.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\uba74 \\(\\left\\{b_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c (1)\uc5d0 \uc758\ud558\uc5ec \uc5b4\ub5a4 \uc2e4\uc218 \\(M\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc815\ub9ac 1.3.1\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\na_n=-b_n\\rightarrow-M.<br \/>\n\\]<br \/>\n\ub610\ud55c \\(-M\\)\uc740 \\(\\{a_n\\}\\)\uc758 \ucd5c\ub300\ud558\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc870\uac10\uc18c\ud558\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc778 \uc218\uc5f4\ub3c4 \uc790\uc2e0\uc758 \ucd5c\ub300\ud558\uacc4\uc5d0 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub294 \uadf9\ud55c\uac12\uc744 \ubbf8\ub9ac \uc54c\uc9c0 \ubabb\ud558\ub294 \uc0c1\ud0dc\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834\uc744 \uc99d\uba85\ud560 \ub54c \ud2b9\ud788 \uc720\uc6a9\ud558\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\\[<br \/>\na_1=1,<br \/>\n\\qquad<br \/>\na_{n+1}=\\sqrt{2+a_n}.<br \/>\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><br \/>\n\uba3c\uc800 \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n1\\le a_n\\le2<br \/>\n\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc790. \\(a_1=1\\)\uc774\ubbc0\ub85c \\(n=1\\)\uc5d0\uc11c\ub294 \uc131\ub9bd\ud55c\ub2e4. \uc5b4\ub5a4 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n1\\le a_n\\le2<br \/>\n\\]<br \/>\n\ub77c\uace0 \uac00\uc815\ud558\uba74<br \/>\n\\[<br \/>\n1<br \/>\n<\n\\sqrt{3}\n\\le\na_{n+1}\n=\n\\sqrt{2+a_n}\n\\le\n\\sqrt4\n=\n2.\n\\]\n\ub530\ub77c\uc11c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n1\\le a_n\\le2.\n\\]\n\ud2b9\ud788 \\(\\left\\{a_n\\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\ub610\ud55c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\na_{n+1}^2-a_n^2<br \/>\n&#038;=<br \/>\n2+a_n-a_n^2\\\\<br \/>\n&#038;=<br \/>\n(2-a_n)(a_n+1)\\\\<br \/>\n&#038;\\ge0.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\\(a_n\\)\uacfc \\(a_{n+1}\\)\uc774 \ubaa8\ub450 \uc591\uc218\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\na_{n+1}\\ge a_n.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\left\\{a_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{a_n\\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uc218\uc5f4\uc5d0\uc11c \uc720\ud55c \uac1c\uc758 \ucc98\uc74c \ud56d\uc744 \uc81c\uac70\ud574\ub3c4 \uadf9\ud55c\uc740 \ubcc0\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\lim_{n\\rightarrow\\infty}a_{n+1}=L.<br \/>\n\\]<br \/>\n\uc815\ub9ac 1.3.1\uc744 \uc774\uc6a9\ud558\uba74<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nL<br \/>\n&#038;=<br \/>\n\\lim_{n\\rightarrow\\infty}a_{n+1}\\\\<br \/>\n&#038;=<br \/>\n\\lim_{n\\rightarrow\\infty}\\sqrt{2+a_n}\\\\<br \/>\n&#038;=<br \/>\n\\sqrt{<br \/>\n2+\\lim_{n\\rightarrow\\infty}a_n<br \/>\n}\\\\<br \/>\n&#038;=<br \/>\n\\sqrt{2+L}.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nL^2=L+2,<br \/>\n\\]<br \/>\n\uc989<br \/>\n\\[<br \/>\n(L-2)(L+1)=0.<br \/>\n\\]<br \/>\n\ubaa8\ub4e0 \\(a_n\\ge1\\)\uc774\ubbc0\ub85c \uc815\ub9ac 1.3.2\uc5d0 \uc758\ud558\uc5ec \\(L\\ge1\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nL=2.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\lim_{n\\rightarrow\\infty}a_n=2.<br \/>\n\\]\n<\/p>\n<\/div>\n<p>\uc218\uc5f4\uc774 \uadc0\ub0a9\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c\uc5d0\ub294 \uba3c\uc800 \uc218\uc5f4\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \ud655\uc778\ud55c \ub4a4 \uadf9\ud55c\ubc95\uce59\uc744 \uc801\uc6a9\ud574\uc57c \ud55c\ub2e4. \uc218\ub834 \uc5ec\ubd80\ub97c \ud655\uc778\ud558\uc9c0 \uc54a\uace0 \uadf9\ud55c\uac12\uc5d0 \uad00\ud55c \ubc29\uc815\uc2dd\ub9cc \uc138\uc6b0\uba74 \uc798\ubabb\ub41c \uacb0\ub860\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/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<br \/>\n\\[<br \/>\nb_1=b_2=1,<br \/>\n\\qquad<br \/>\nb_{n+2}=b_{n+1}+b_n<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n<p>\ub9cc\uc57d \\(\\left\\{b_n\\right\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0 \uadf8 \uadf9\ud55c\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uba74<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nL<br \/>\n&#038;=<br \/>\n\\lim_{n\\rightarrow\\infty}b_{n+2}\\\\<br \/>\n&#038;=<br \/>\n\\lim_{n\\rightarrow\\infty}(b_{n+1}+b_n)\\\\<br \/>\n&#038;=<br \/>\nL+L<br \/>\n=<br \/>\n2L.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(L=0\\)\uc744 \uc5bb\uac8c \ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc774\uac83\uc740 \uc218\uc5f4\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uc798\ubabb\ub41c \uac00\uc815\uc5d0\uc11c \uc5bb\uc740 \uacb0\ub860\uc774\ub2e4.<\/p>\n<p>\uc2e4\uc81c\ub85c \ubaa8\ub4e0 \ud56d\uc740 \\(1\\) \uc774\uc0c1\uc774\uace0, \\(n\\ge2\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nb_{n+1}<br \/>\n=<br \/>\nb_n+b_{n-1}<br \/>\n\\ge<br \/>\nb_n+1.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \uadc0\ub0a9\uc801\uc73c\ub85c<br \/>\n\\[<br \/>\nb_n\\ge n-1<br \/>\n\\qquad(n\\ge2)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[<br \/>\nn-1\\rightarrow\\infty<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\nb_n\\rightarrow\\infty.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\left\\{b_n\\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\uace0 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/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 \\(E\\)\uac00 \uc2e4\uc218\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\( x\\le M \\) \uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uba74 \\(E\\)\uac00 \uc704\ub85c \uc720\uacc4(bounded above)\ub77c\uace0 \ub9d0\ud558\uace0 \\(M\\)\uc744 \\(E\\)\uc758 \uc0c1\uacc4\ub77c\uace0 \ubd80\ub978\ub2e4. \ubc18\ub300\ub85c \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\( m\\le x \\) \ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud558\uba74 \\(E\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4(bounded below)\ub77c\uace0 \ub9d0\ud558\uace0 \\(m\\)\uc744 \\(E\\)\uc758 \ud558\uacc4(lower bound)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9d1\ud569 \\(E\\)\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":"no","_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":16,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6675\/revisions"}],"predecessor-version":[{"id":10135,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6675\/revisions\/10135"}],"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}]}}