{"id":6677,"date":"2021-07-20T23:50:15","date_gmt":"2021-07-20T14:50:15","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6677"},"modified":"2021-07-22T22:48:08","modified_gmt":"2021-07-22T13:48:08","slug":"limit-superior-and-limit-inferior","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/limit-superior-and-limit-inferior\/","title":{"rendered":"\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 6\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \\(a_n = (-1)^n\\)\uc73c\ub85c \uc815\uc758\ub41c \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uc218\uc5f4\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc774 \uc218\uc5f4\uc758 \ub450 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ a_{2n}\\right\\}\\)\uacfc \\(\\left\\{a_{2n+1}\\right\\}\\)\uc744 \uc0b4\ud3b4\ubcf4\uba74 \\(n\\,\\rightarrow\\,\\infty\\)\uc77c \ub54c
\n\\[a_{2n} \\,\\rightarrow\\,1 \\quad\\text{and}\\quad a_{2n+1} \\,\\rightarrow\\,-1\\]
\n\ub85c\uc11c \uac01\uac01 \\(1\\)\uacfc \\(-1\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \ub354\uc6b1\uc774, \ub9cc\uc57d \\(\\left\\{a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\uace0 \uc5b4\ub5a4 \uac12\uc5d0 \uc218\ub834\ud558\ub2e4\uba74 \uadf8 \uadf9\ud55c\uac12\uc740 \\(1\\)\ubcf4\ub2e4 \ud074 \uc218 \uc5c6\uace0 \\(-1\\)\ubcf4\ub2e4 \uc791\uc744 \uc218 \uc5c6\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(1\\)\uacfc \\(-1\\)\uc744 \uac01\uac01 \\(\\left\\{a_n\\right\\}\\)\uc758 \uc0c1\uadf9\ud55c, \ud558\uadf9\ud55c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc2e4\uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc774 \uac1c\ub150\uc744 \ub354 \uc815\ud655\ud558\uac8c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

<\/p>\n

\uc218\uc5f4\uc758 \uc9d1\uc801\uc810<\/h2>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0 \\(\\lambda\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\left\\{a_n\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{a_{r_n}\\right\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc774 \ubd80\ubd84\uc218\uc5f4\uc774 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(\\lambda\\)\ub97c \\(\\left\\{a_n\\right\\}\\)\uc758 \uc9d1\uc801\uc810<\/span>(cluster point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\ubcf4\uae30 1.6.1.<\/span><\/p>\n

    \n
  1. \uc218\uc5f4 \\(\\left\\{ \\frac{1}{n}\\right\\}\\)\uc740 \\(0\\)\uc744 \uc9d1\uc801\uc810\uc73c\ub85c \uac00\uc9c4\ub2e4. \\(0\\)\uc740 \uc774 \uc218\uc5f4\uc758 \uc720\uc77c\ud55c \uc9d1\uc801\uc810\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\left\\{ \\frac{1}{n}\\right\\}\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc774 \uc218\uc5f4\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc218\uc5f4\uc740 \\(0\\) \uc774\uc678\uc758 \ub2e4\ub978 \uac12\uc5d0 \uc218\ub834\ud560 \uc218 \uc5c6\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n
  2. \uc591\uc758 \uc815\uc218 \\(n\\)\uc758 100\uc758 \uc790\ub9ac \uc22b\uc790\ub97c \\(h_n\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e8 \\(n < 100\\)\uc77c \ub54c\ub294 \\(h_n = 0\\)\uc73c\ub85c \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(0\\)\ubd80\ud130 \\(9\\)\uae4c\uc9c0\uc758 \ubaa8\ub4e0 \uc815\uc218\ub294 \uc218\uc5f4 \\(\\left\\{ h_n \\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.<\/li>\n
  3. \uad70\uc218\uc5f4
    \n\\[1,\\,1,\\,2,\\,1,\\,2,\\,3,\\,1,\\,2,\\,3,\\,4,\\,1,\\,2,\\,3,\\,4,\\,5,\\,\\cdots\\]
    \n\uc740 \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218\ub97c \uc9d1\uc801\uc810\uc73c\ub85c \uac00\uc9c4\ub2e4.\n<\/li>\n
  4. \ud568\uc218 \\(\\phi : \\mathbb{N} \\rightarrow \\mathbb{Q}\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc774\ub77c\uace0 \ud558\uace0, \\(r_n = \\phi (n)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc2e4\uc218\ub294 \\(\\left\\{ r_n \\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc591\uc218 \\(M\\)\uc774 \\(\\left\\lvert a_n \\right\\rvert\\)\uc758 \uc0c1\uacc4\ub77c\uace0 \ud558\uc790. \uc989 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\( \\left\\lvert a_n \\right\\rvert \\le M\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \\(I = [-M, \\,M]\\)\uc774\ub77c\uace0 \ud558\uba74 \uc774 \ub2eb\ud78c\uad6c\uac04\uc744 \uae38\uc774\uac00 \uac19\uc740 \ub450 \ub2eb\ud78c\uad6c\uac04\uc73c\ub85c \ucabc\uac20 \\([-M ,\\, 0]\\)\uacfc \\([0,\\,M]\\) \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(\\left\\{a_n\\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c4\ub2e4. \uadf8 \uad6c\uac04\uc744 \\(I_0 = \\left[x_0 ,\\, y_0 \\right]\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc2dc \\(I_0\\)\ub97c \ub450 \ub2eb\ud78c\uad6c\uac04\uc73c\ub85c \ucabc\uac20
    \n\\[\\left[x_0 ,\\, \\frac{x_0 + y_0}{2} \\right] \\quad\\text{and}\\quad \\left[\\frac{x_0 + y_0}{2} ,\\, y_0 \\right]\\]
    \n\uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(\\left\\{a_n\\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c4\ub2e4. \uadf8 \uad6c\uac04\uc744 \\(I_1 = \\left[x_1 ,\\, y_1 \\right]\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74 \ub2eb\ud78c\uad6c\uac04\uc5f4
    \n\\[I_0 ,\\, I_1 ,\\, I_2 ,\\, \\cdots\\]
    \n\ub97c \uc5bb\ub294\ub2e4. \\(I_n = \\left[ x_n ,\\, y_n \\right]\\)\uc774\ub77c\uace0 \ud558\uc790. \uad6c\uac04 \\(I_n\\)\uc758 \uae38\uc774\ub294 \\(2^{-n} M\\)\uc774\uba70 \\(\\left\\{a_n\\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9c4\ub2e4.<\/p>\n

    \\(I_1\\)\uc5d0 \uc18d\ud558\ub294 \ud56d\uc744 \ud558\ub098 \uc7a1\uc544\uc11c \\(a_{m_1} \\in I_1\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc73c\ub85c
    \n\\[a_{m_2}\\in I_2 \\quad\\text{and}\\quad m_1 < m_2 \\]\n\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc815\uc218 \\(m_2\\)\ub97c \ud0dd\ud558\uc790. \uadc0\ub0a9\uc801\uc73c\ub85c, \uc815\uc218 \\(m_k\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c\n\\[a_{m_{k+1}}\\in I_{k+1} \\quad\\text{and}\\quad m_k < m_{k+1} \\]\n\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc815\uc218 \\(m_{k+1}\\)\uc744 \ud0dd\ud558\uc790.<\/p>\n

    \\(I_n\\)\uc758 \uc67c\ucabd \ub05d\uc810\uc73c\ub85c \ub9cc\ub4e0 \uc218\uc5f4 \\(\\left\\{x_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0, \\(I_n\\)\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc73c\ub85c \ub9cc\ub4e0 \uc218\uc5f4 \\(\\left\\{y_n\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \ub610\ud55c \uc774 \ub450 \uc218\uc5f4\uc740 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc774 \ub450 \uc218\uc5f4\uc740 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uac12\uc744 \uac01\uac01 \\(X,\\) \\(Y\\)\ub77c \ud558\uc790. \\(n\\,\\rightarrow\\,\\infty\\)\uc77c \ub54c \uad6c\uac04 \\(I_n\\)\uc758 \uae38\uc774\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(\\left\\lvert x_n – y_n \\right\\rvert \\,\\rightarrow\\,0\\)\uc774\ub2e4. \uc989 \\(\\left\\{x_n\\right\\}\\)\uacfc \\(\\left\\{y_n\\right\\}\\)\uc740 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(X=Y\\)\uc774\ub2e4. \uc774 \uac12\uc744 \\(\\lambda = X = Y\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[x_n \\le a_{m_n} \\le y_n\\]
    \n\uc774\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{a_{m_n}\\right\\}\\)\uc740 \\(\\lambda\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\left\\{a_n\\right\\}\\)\uc740 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

    \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1.6.1. (Bolzano-Weierstrass \uc815\ub9ac)<\/span><\/p>\n

    \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\uba74 \\(\\left\\{a_n\\right\\}\\)\uc740 \uc9d1\uc801\uc810\uc744 \uac00\uc9c4\ub2e4.\n<\/p>\n<\/div>\n

    \ucc38\uace0.<\/span> \uc704 \uc815\ub9ac\uc758 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300
    \n\\[r_n = (-2)^n + 2^n\\]
    \n\uc774\ub77c\uace0 \ud558\uba74 \uc218\uc5f4 \\(\\left\\{ r_n \\right\\}\\)\uc740 \\(0\\)\uc744 \uc9d1\uc801\uc810\uc73c\ub85c \uac16\uc9c0\ub9cc \\(\\left\\{r_n\\right\\}\\)\uc740 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.
    \n<\/span><\/p>\n

    <\/p>\n

    \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c<\/h2>\n

    \\(\\left\\{a_n \\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\left\\{a_n \\right\\}\\)\uc758 \uc9d1\uc801\uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(C\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(C\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \uc774\ub54c \\(C\\)\uc758 \ucd5c\ub313\uac12\uc744 \\(\\left\\{a_n \\right\\}\\)\uc758 \uc0c1\uadf9\ud55c<\/span>(limit superior)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \\(C\\)\uc758 \ucd5c\uc19f\uac12\uc744 \\(\\left\\{a_n \\right\\}\\)\uc758 \ud558\uadf9\ud55c<\/span>(limit inferior)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \\(\\left\\{a_n\\right\\}\\)\uc758 \uc0c1\uadf9\ud55c\uc774 \\(\\overline{L}\\)\uc778 \uac83\uc744 \uae30\ud638\ub85c
    \n\\[\\varlimsup _{n\\rightarrow\\infty} a_n = \\overline{L}\\]
    \n\ub610\ub294
    \n\\[\\limsup _{n\\rightarrow\\infty} a_n = \\overline{L}\\]
    \n\uacfc \uac19\uc774 \ub098\ud0c0\ub0b4\uba70, \\(\\left\\{a_n\\right\\}\\)\uc758 \ud558\uadf9\ud55c\uc774 \\(\\underline{L}\\)\uc778 \uac83\uc744 \uae30\ud638\ub85c
    \n\\[\\varliminf _{n\\rightarrow\\infty} a_n = \\underline{L}\\]
    \n\ub610\ub294
    \n\\[\\liminf _{n\\rightarrow\\infty} a_n = \\underline{L}\\]
    \n\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.\n<\/p>\n

    \uc720\uacc4\uac00 \uc544\ub2cc \uc2e4\uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n