{"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":"2026-09-27T17:16:22","modified_gmt":"2026-09-27T08:16:22","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
\n\\[
\na_n=(-1)^n
\n\\]
\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 \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uacfc \ud640\uc218 \ubc88\uc9f8 \ud56d\ub9cc \uac01\uac01 \uace8\ub77c \ub9cc\ub4e0 \uc218\uc5f4\uc744 \uc0b4\ud3b4\ubcf4\uba74
\n\\[
\na_{2n}\\rightarrow1,
\n\\qquad
\na_{2n+1}\\rightarrow-1
\n\\]
\n\uc774\ub2e4. \uc774\ucc98\ub7fc \uc6d0\ub798 \uc218\uc5f4\uc5d0\uc11c \uc77c\ubd80 \ud56d\uc744 \uc21c\uc11c\ub97c \uc720\uc9c0\ud558\uba74\uc11c \uace8\ub77c \ub9cc\ub4e0 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc870\uc0ac\ud558\uba74, \uc218\ub834\ud558\uc9c0 \uc54a\ub294 \uc218\uc5f4\uc758 \uc7a5\uae30\uc801\uc778 \uc6c0\uc9c1\uc784\uc744 \ub354 \uc790\uc138\ud788 \uc0b4\ud3b4\ubcfc \uc218 \uc788\ub2e4.<\/p>\n

<\/p>\n

\ubd80\ubd84\uc218\uc5f4\uacfc \uc9d1\uc801\uc810<\/h2>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc218\uc5f4\uc758 \ucca8\uc790 \uac00\uc6b4\ub370
\n\\[
\nr_1 < r_2 < r_3 < \\cdots\n\\]\n\uc778 \ucca8\uc790\ub4e4\uc744 \uace8\ub77c \ub9cc\ub4e0 \uc218\uc5f4\n\\[\na_{r_1},\\,a_{r_2},\\,a_{r_3},\\,\\ldots\n\\]\n\uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4<\/span>(subsequence)\uc774\ub77c\uace0 \ubd80\ub974\uace0
\n\\[
\n\\left\\{a_{r_k}\\right\\}
\n\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\\(\\lambda\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \\(\\left\\{a_n\\right\\}\\)\uc758 \uc5b4\ub5a4 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{a_{r_k}\\right\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[
\na_{r_k}\\rightarrow\\lambda
\n\\]
\n\uc774\uba74 \\(\\lambda\\)\ub97c \\(\\left\\{a_n\\right\\}\\)\uc758 \uc9d1\uc801\uc810<\/span>(cluster point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \ubd80\ubd84\uc218\uc5f4\uc740 \ubaa8\ub450 \uc6d0\ub798 \uc218\uc5f4\uacfc \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc2e4\uc81c\ub85c
\n\\[
\na_n\\rightarrow L
\n\\]
\n\uc774\uace0 \\(\\left\\{a_{r_k}\\right\\}\\)\uc774 \ubd80\ubd84\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\epsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
\n\\[
\n|a_n-L| < \\epsilon\n\\]\n\uc774\ub2e4. \ud55c\ud3b8 \\(r_k\\rightarrow\\infty\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0\uc11c\ub294 \\(r_k\\)\ub3c4 \uc774 \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4. \ub530\ub77c\uc11c\n\\[\na_{r_k}\\rightarrow L.\n\\]\n<\/p>\n

\n

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

    \n
  1. \uc218\uc5f4
    \n\\[
    \n\\left\\{\\frac{1}{n}\\right\\}
    \n\\]
    \n\uc740 \\(0\\)\uc744 \uc9d1\uc801\uc810\uc73c\ub85c \uac00\uc9c4\ub2e4. \ub610\ud55c \uc774 \uc218\uc5f4\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \ubaa8\ub4e0 \ubd80\ubd84\uc218\uc5f4\ub3c4 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \\(0\\)\uc740 \uc774 \uc218\uc5f4\uc758 \uc720\uc77c\ud55c \uc9d1\uc801\uc810\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\uc5d0\ub294 \\(h_n=0\\)\uc73c\ub85c \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(0,1,\\ldots,9\\)\uac00 \uac01\uac01 \ubb34\ud55c\ud788 \ub9ce\uc774 \ub098\ud0c0\ub098\ubbc0\ub85c \uc774\ub4e4 \ubaa8\ub450\uac00 \\(\\left\\{h_n\\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.<\/li>\n
  3. \uc218\uc5f4
    \n\\[
    \n1,\\,1,\\,2,\\,1,\\,2,\\,3,\\,1,\\,2,\\,3,\\,4,\\,1,\\,2,\\,3,\\,4,\\,5,\\,\\cdots
    \n\\]
    \n\uc740 \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218\ub97c \uc9d1\uc801\uc810\uc73c\ub85c \uac00\uc9c4\ub2e4.<\/li>\n
  4. \ud568\uc218
    \n\\[
    \n\\phi:\\mathbb{N}\\rightarrow\\mathbb{Q}
    \n\\]
    \n\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc774\uace0
    \n\\[
    \nr_n=\\phi(n)
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uc720\ub9ac\uc218\uac00 \uc2e4\uc218 \uc548\uc5d0\uc11c \uc870\ubc00\ud558\ubbc0\ub85c \uc784\uc758\uc758 \uc2e4\uc218 \\(\\lambda\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c0\ub294 \uc11c\ub85c \ub2e4\ub978 \uc720\ub9ac\uc218\ub4e4\uc744 \ucc28\ub840\ub85c \uace0\ub97c \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \uc784\uc758\uc758 \uc2e4\uc218\ub294 \\(\\left\\{r_n\\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

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

    \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc740 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \uc9d1\uc801\uc810\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85.<\/p>\n

    \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5b4\ub5a4 \uc591\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|a_n|\\le M
    \n\\]
    \n\uc774\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \ud56d\uc740 \ub2eb\ud78c\uad6c\uac04
    \n\\[
    \nI_0=[-M,M]
    \n\\]
    \n\uc5d0 \uc18d\ud55c\ub2e4.<\/p>\n

    \\(I_0\\)\ub97c \uae38\uc774\uac00 \uac19\uc740 \ub450 \ub2eb\ud78c\uad6c\uac04\uc73c\ub85c \ub098\ub204\uba74 \uadf8\uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uc5d0\ub294 \uc218\uc5f4\uc758 \ud56d\uc774 \ubb34\ud55c\ud788 \ub9ce\uc774 \ub4e4\uc5b4 \uc788\ub2e4. \uadf8\ub7ec\ud55c \uad6c\uac04 \ud558\ub098\ub97c \\(I_1\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc2dc \\(I_1\\)\uc744 \uae38\uc774\uac00 \uac19\uc740 \ub450 \ub2eb\ud78c\uad6c\uac04\uc73c\ub85c \ub098\ub204\uace0, \uc218\uc5f4\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \ud3ec\ud568\ud558\ub294 \uad6c\uac04 \ud558\ub098\ub97c \\(I_2\\)\ub77c\uace0 \ud558\uc790. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec
    \n\\[
    \nI_0\\supseteq I_1\\supseteq I_2\\supseteq\\cdots
    \n\\]
    \n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n\\[
    \nI_k=[x_k,y_k]
    \n\\]
    \n\ub77c\uace0 \ud558\uba74 \\(I_k\\)\ub294 \uc218\uc5f4\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \ud3ec\ud568\ud558\uace0 \uadf8 \uae38\uc774\ub294
    \n\\[
    \ny_k-x_k=\\frac{2M}{2^k}
    \n\\]
    \n\uc774\ub2e4.<\/p>\n

    \uac01 \\(I_k\\)\uac00 \uc218\uc5f4\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \ud3ec\ud568\ud558\ubbc0\ub85c \ucca8\uc790\ub4e4\uc744
    \n\\[
    \nm_1 < m_2 < m_3 < \\cdots\n\\]\n\uac00 \ub418\ub3c4\ub85d \ud0dd\ud558\uba74\uc11c\n\\[\na_{m_k}\\in I_k\n\\]\n\uac00 \ub418\uac8c \ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(\\left\\{a_{m_k}\\right\\}\\)\uc740 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774\ub2e4.<\/p>\n

    \ud55c\ud3b8 \\(\\left\\{x_k\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\uc774\uace0, \\(\\left\\{y_k\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uba74\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc5b4\ub5a4 \uc2e4\uc218 \\(X,Y\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nx_k\\rightarrow X,
    \n\\qquad
    \ny_k\\rightarrow Y.
    \n\\]
    \n\uadf8\ub7f0\ub370
    \n\\[
    \ny_k-x_k=\\frac{2M}{2^k}\\rightarrow0
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \nY-X=0.
    \n\\]
    \n\ub530\ub77c\uc11c \\(X=Y\\)\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744 \\(\\lambda\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

    \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nx_k\\le a_{m_k}\\le y_k
    \n\\]
    \n\uc774\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \na_{m_k}\\rightarrow\\lambda.
    \n\\]
    \n\ub530\ub77c\uc11c \\(\\lambda\\)\ub294 \\(\\left\\{a_n\\right\\}\\)\uc758 \uc9d1\uc801\uc810\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ucc38\uace0.<\/span> Bolzano-Weierstrass \uc815\ub9ac\uc758 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4
    \n\\[
    \nr_n=(-2)^n+2^n
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \ud640\uc218 \\(n\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(r_n=0\\)\uc774\ubbc0\ub85c \\(0\\)\uc740 \uc774 \uc218\uc5f4\uc758 \uc9d1\uc801\uc810\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uc740
    \n\\[
    \nr_{2n}=2^{2n+1}
    \n\\]
    \n\uc774\uace0 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ubbc0\ub85c \\(\\left\\{r_n\\right\\}\\)\uc740 \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/p>\n

    <\/p>\n

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

    \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc744 \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uc218\uc5f4\uc758 \ub4b7\ubd80\ubd84\uc5d0 \ub098\ud0c0\ub098\ub294 \uac12\ub4e4\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0 \\(n\\)\uc774 \ucca8\uc790\ub77c\uace0 \ud558\uc790. \\(n\\)\ubc88\uc9f8 \uc774\ud6c4\uc758 \ud56d\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569\uc744
    \n\\[
    \nE_n
    \n=
    \n\\left\\{
    \na_k
    \n\\,\\middle|\\,
    \nk\\ge n
    \n\\right\\}
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \\(\\infty\\)\uc640 \\(-\\infty\\)\uae4c\uc9c0 \ud5c8\uc6a9\ud558\uc5ec \uc0dd\uac01\ud558\uae30\ub85c \ud55c\ub2e4. \uc989 \uc9d1\ud569\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \uadf8 \uc0c1\ud55c\uc744 \\(\\infty\\), \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \uadf8 \ud558\ud55c\uc744 \\(-\\infty\\)\ub77c\uace0 \uc4f0\uae30\ub85c \ud55c\ub2e4.<\/p>\n

    \uac01 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \ns_n=\\sup E_n,
    \n\\qquad
    \nt_n=\\inf E_n
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \\(E_{n+1}\\subseteq E_n\\)\uc774\ubbc0\ub85c
    \n\\[
    \ns_{n+1}\\le s_n,
    \n\\qquad
    \nt_n\\le t_{n+1}.
    \n\\]
    \n\uc989 \\(\\left\\{s_n\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \\(\\left\\{t_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/p>\n

    \\(\\left\\{a_n\\right\\}\\)\uc758 \uc0c1\uadf9\ud55c<\/span>(limit superior)\uacfc \ud558\uadf9\ud55c<\/span>(limit inferior)\uc744 \uac01\uac01
    \n\\[
    \n\\varlimsup_{n\\rightarrow\\infty}a_n
    \n=
    \n\\inf_n\\sup_{k\\ge n}a_k
    \n\\]
    \n\ubc0f
    \n\\[
    \n\\varliminf_{n\\rightarrow\\infty}a_n
    \n=
    \n\\sup_n\\inf_{k\\ge n}a_k
    \n\\]
    \n\ub85c \uc815\uc758\ud55c\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f0\uae30\ub3c4 \ud55c\ub2e4.
    \n\\[
    \n\\limsup_{n\\rightarrow\\infty}a_n,
    \n\\qquad
    \n\\liminf_{n\\rightarrow\\infty}a_n.
    \n\\]
    \n\uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc758 \uac12\uc73c\ub85c\ub294 \uc2e4\uc218\ubfd0 \uc544\ub2c8\ub77c \\(\\infty\\)\uc640 \\(-\\infty\\)\ub3c4 \ud5c8\uc6a9\ud55c\ub2e4.<\/p>\n

    \ub530\ub77c\uc11c \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc740 \ubaa8\ub4e0 \uc2e4\uc218\uc5f4\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uc218\uc5f4\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \ubaa8\ub4e0 \uaf2c\ub9ac\uc9d1\ud569 \\(E_n\\)\ub3c4 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c
    \n\\[
    \n\\varlimsup_{n\\rightarrow\\infty}a_n=\\infty.
    \n\\]
    \n\ub9c8\ucc2c\uac00\uc9c0\ub85c \uc218\uc5f4\uc774 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74
    \n\\[
    \n\\varliminf_{n\\rightarrow\\infty}a_n=-\\infty.
    \n\\]
    \n\ub610\ud55c
    \n\\[
    \na_n\\rightarrow\\infty
    \n\\]
    \n\uc774\uba74
    \n\\[
    \n\\varliminf_{n\\rightarrow\\infty}a_n
    \n=
    \n\\varlimsup_{n\\rightarrow\\infty}a_n
    \n=
    \n\\infty,
    \n\\]
    \n\uc774\uace0
    \n\\[
    \na_n\\rightarrow-\\infty
    \n\\]
    \n\uc774\uba74 \ub450 \uac12\uc774 \ubaa8\ub450 \\(-\\infty\\)\uc774\ub2e4.<\/p>\n

    \n

    \ubcf4\uae30 1.6.2.<\/span><\/p>\n

      \n
    1. \ub9cc\uc57d \\(a_n=(-1)^n\\)\uc774\uba74
      \n\\[
      \n\\varlimsup_{n\\rightarrow\\infty}a_n=1,
      \n\\qquad
      \n\\varliminf_{n\\rightarrow\\infty}a_n=-1.
      \n\\]\n<\/li>\n
    2. \ub9cc\uc57d
      \n\\[
      \nb_n=(-2)^n+2^n
      \n\\]
      \n\uc774\uba74 \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0 \ud640\uc218 \ubc88\uc9f8 \ud56d\uc740 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c
      \n\\[
      \n\\varlimsup_{n\\rightarrow\\infty}b_n=\\infty,
      \n\\qquad
      \n\\varliminf_{n\\rightarrow\\infty}b_n=0.
      \n\\]\n<\/li>\n
    3. \ub9cc\uc57d \\(c_n=(-2)^n\\)\uc774\uba74
      \n\\[
      \n\\varlimsup_{n\\rightarrow\\infty}c_n=\\infty,
      \n\\qquad
      \n\\varliminf_{n\\rightarrow\\infty}c_n=-\\infty.
      \n\\]\n<\/li>\n
    4. \ub9cc\uc57d
      \n\\[
      \nd_n=\\left(\\frac12\\right)^n
      \n\\]
      \n\uc774\uba74 \\(d_n\\rightarrow0\\)\uc774\ubbc0\ub85c
      \n\\[
      \n\\varlimsup_{n\\rightarrow\\infty}d_n
      \n=
      \n\\varliminf_{n\\rightarrow\\infty}d_n
      \n=
      \n0.
      \n\\]\n<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc815\ub9ac 1.6.2.<\/span><\/p>\n

      \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\uace0 \uc9d1\uc801\uc810\ub4e4\uc758 \uc9d1\ud569\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. \ub610\ud55c
      \n\\[
      \n\\max C
      \n=
      \n\\varlimsup_{n\\rightarrow\\infty}a_n
      \n\\]
      \n\uc774\uace0
      \n\\[
      \n\\min C
      \n=
      \n\\varliminf_{n\\rightarrow\\infty}a_n.
      \n\\]
      \n\ud2b9\ud788
      \n\\[
      \n\\varliminf_{n\\rightarrow\\infty}a_n
      \n\\le
      \n\\varlimsup_{n\\rightarrow\\infty}a_n.
      \n\\]\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85.<\/p>\n

      Bolzano-Weierstrass \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(C\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\uc790.
      \n\\[
      \ns_n=\\sup_{k\\ge n}a_k,
      \n\\qquad
      \nt_n=\\inf_{k\\ge n}a_k.
      \n\\]
      \n\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc774\ubbc0\ub85c \\(s_n\\)\uacfc \\(t_n\\)\uc740 \ubaa8\ub450 \uc2e4\uc218\uc774\ub2e4. \ub610\ud55c \\(\\left\\{s_n\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba70, \\(\\left\\{t_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \ns_n\\rightarrow S,
      \n\\qquad
      \nt_n\\rightarrow T
      \n\\]
      \n\uc778 \uc2e4\uc218 \\(S,T\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc815\uc758\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \nS=\\varlimsup_{n\\rightarrow\\infty}a_n,
      \n\\qquad
      \nT=\\varliminf_{n\\rightarrow\\infty}a_n.
      \n\\]\n<\/p>\n

      \uba3c\uc800 \\(S\\)\uac00 \uc9d1\uc801\uc810\uc784\uc744 \ubcf4\uc774\uc790. \\(n_1 < n_2 < \\cdots\\)\ub97c \ucda9\ubd84\ud788 \ube60\ub974\uac8c \uc99d\uac00\ud558\ub3c4\ub85d \ud0dd\ud558\uace0, \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\na_{m_j}>s_{n_j}-\\frac1j,
      \n\\qquad
      \nm_j\\ge n_j
      \n\\]
      \n\uc774\uba74\uc11c
      \n\\[
      \nm_1 < m_2 < \\cdots\n\\]\n\uac00 \ub418\ub3c4\ub85d \\(m_j\\)\ub97c \uace0\ub97c \uc218 \uc788\ub2e4. \uc0c1\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc774\ub7ec\ud55c \ud56d\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c\n\\[\na_{m_j}\\le s_{n_j}.\n\\]\n\ub530\ub77c\uc11c\n\\[\ns_{n_j}-\\frac1j\n<\na_{m_j}\n\\le\ns_{n_j}.\n\\]\n\\(s_{n_j}\\rightarrow S\\)\uc774\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec\n\\[\na_{m_j}\\rightarrow S.\n\\]\n\ub530\ub77c\uc11c \\(S\\in C\\)\uc774\ub2e4.<\/p>\n

      \uc774\uc81c \\(\\lambda\\in C\\)\ub77c\uace0 \ud558\uc790. \uc5b4\ub5a4 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{a_{r_j}\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \na_{r_j}\\rightarrow\\lambda
      \n\\]
      \n\uc774\ub2e4. \uc784\uc758\uc758 \uace0\uc815\ub41c \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(j\\)\uc5d0\uc11c\ub294 \\(r_j\\ge n\\)\uc774\ubbc0\ub85c
      \n\\[
      \na_{r_j}\\le s_n.
      \n\\]
      \n\uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\lambda\\le s_n.
      \n\\]
      \n\uc774\ub294 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c
      \n\\[
      \n\\lambda\\le S.
      \n\\]
      \n\ub530\ub77c\uc11c \\(S\\)\ub294 \\(C\\)\uc758 \ucd5c\ub313\uac12\uc774\ub2e4.<\/p>\n

      \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(T\\)\uc5d0 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \ub9cc\ub4e4 \uc218 \uc788\uace0, \ubaa8\ub4e0 \\(\\lambda\\in C\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \nT\\le\\lambda
      \n\\]
      \n\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c
      \n\\[
      \nT=\\min C.
      \n\\]
      \n\uadf8\ub7ec\ubbc0\ub85c
      \n\\[
      \n\\varliminf_{n\\rightarrow\\infty}a_n
      \n=
      \nT
      \n\\le
      \nS
      \n=
      \n\\varlimsup_{n\\rightarrow\\infty}a_n.
      \n\\]
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 1.6.3.<\/span><\/p>\n

      \\(\\left\\{a_n\\right\\}\\)\uc774 \uc720\uacc4\uc778 \uc2e4\uc218\uc5f4\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ub450 \uc870\uac74\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n

        \n
      1. \\(\\left\\{a_n\\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
      2. \n\\[
        \n\\varliminf_{n\\rightarrow\\infty}a_n
        \n=
        \n\\varlimsup_{n\\rightarrow\\infty}a_n
        \n=
        \nL.
        \n\\]\n<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85.<\/p>\n

        \\(\\left\\{a_n\\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc758 \ubaa8\ub4e0 \ubd80\ubd84\uc218\uc5f4\ub3c4 \\(L\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(L\\)\uc740 \uc720\uc77c\ud55c \uc9d1\uc801\uc810\uc774\ub2e4. \uc815\ub9ac 1.6.2\uc5d0 \uc758\ud558\uc5ec
        \n\\[
        \n\\varliminf_{n\\rightarrow\\infty}a_n
        \n=
        \n\\varlimsup_{n\\rightarrow\\infty}a_n
        \n=
        \nL.
        \n\\]\n<\/p>\n

        \uac70\uafb8\ub85c
        \n\\[
        \n\\varliminf_{n\\rightarrow\\infty}a_n
        \n=
        \n\\varlimsup_{n\\rightarrow\\infty}a_n
        \n=
        \nL
        \n\\]
        \n\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\ub294\ub2e4.
        \n\\[
        \ns_n=\\sup_{k\\ge n}a_k,
        \n\\qquad
        \nt_n=\\inf_{k\\ge n}a_k.
        \n\\]
        \n\uadf8\ub7ec\uba74
        \n\\[
        \ns_n\\rightarrow L,
        \n\\qquad
        \nt_n\\rightarrow L.
        \n\\]
        \n\ub610\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[
        \nt_n\\le a_n\\le s_n.
        \n\\]
        \n\ub530\ub77c\uc11c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
        \n\\[
        \na_n\\rightarrow L.
        \n\\]
        \n<\/span><\/p>\n<\/div>\n

        \uc815\ub9ac 1.6.3\uc740 \uc8fc\uc5b4\uc9c4 \uc720\uacc4\uc218\uc5f4\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\uc815\ud560 \ub54c \uc720\uc6a9\ud558\ub2e4. \ud2b9\ud788 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc774 \uc11c\ub85c \ub2e4\ub974\uba74 \uadf8 \uc218\uc5f4\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

        \n

        \ubcf4\uae30 1.6.3.<\/span><\/p>\n

          \n
        1. \\(a_n=(-1)^n\\)\uc774\uba74
          \n\\[
          \n\\varlimsup_{n\\rightarrow\\infty}a_n
          \n=
          \n1
          \n\\ne
          \n-1
          \n=
          \n\\varliminf_{n\\rightarrow\\infty}a_n
          \n\\]
          \n\uc774\ubbc0\ub85c \\(\\left\\{a_n\\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n
        2. \uc218\uc5f4 \\(\\left\\{b_n\\right\\}\\)\uc774
          \n\\[
          \n3,\\,6,\\,9,\\,3,\\,6,\\,9,\\,3,\\,6,\\,9,\\,\\cdots
          \n\\]
          \n\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uba74
          \n\\[
          \n\\varlimsup_{n\\rightarrow\\infty}b_n
          \n=
          \n9
          \n\\ne
          \n3
          \n=
          \n\\varliminf_{n\\rightarrow\\infty}b_n.
          \n\\]
          \n\ub530\ub77c\uc11c \\(\\left\\{b_n\\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

          <\/p>\n

          \n\\[
          \n\\newcommand{\\Hom}{{\\operatorname{Hom}}}
          \n\\newcommand{\\Mat}{{\\operatorname{Mat}}}
          \n\\newcommand{\\proj}{{\\operatorname{proj}}}
          \n\\newcommand{\\adj}{{\\operatorname{adj}}}
          \n\\]\n<\/div>\n
          \n