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People who take cat naps don't usually sleep in a cat's cradle.
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'''163ÂÊ, 178ÂÊ''': ¿À·£¸¸¿¡ ¼ö½Ä Á¶ÆÇ¿¡ µµÀüÇØ º¸¾Ò½À´Ï´Ù. ¸ñÇ¥´Â ´ÙÀ½ ¼ö½ÄÀÔ´Ï´Ù. $$\lim_{n\to\infty}x_n {\rm\ exists} \iff \mathop{\smash\limsup\vphantom\liminf}_{n\to\infty}x_n = \liminf_{n\to\infty}x_n$$ * '''½Ãµµ I''': ¸ÕÀú, ¼øÁøÇÏ°Ô ´«¿¡ º¸ÀÌ´Â ±×´ë·Î ¾Æ·¡¿Í °°ÀÌ ½ÃµµÇغ¸¾Ò½À´Ï´Ù. ¾Æ¹«·± ±â±³µµ ¾øÀÌ. {{{#!vim tex $$ \lim_{n\to\infty}x_n {\rm\ exists} \iff \limsup_{n\to\infty}x_n = \liminf_{n\to\infty}x_n $$ \}}} ±×·¨´õ´Ï, °á°ú ¿ª½Ã ¼øÁøÇÏ°Ô Å¸ÀÌÇÎ ÇÑ ±×´ë·Î º¸¿©ÁÝ´Ï´Ù. $$\lim_{n\to\infty}x_n {\rm\ exists} \iff \limsup_{n\to\infty}x_n = \liminf_{n\to\infty}x_n$$ »ç½Ç ±â´ëµµ ¾ÈÇß½À´Ï´Ù. :( $$n\to\infty$$ ÀÇ À§Ä¡°¡ ÀÏÁ¤ÇÏÁö ¾Ê°í ¼·Î ´Ù¸¨´Ï´Ù. ƯÈ÷ limsup ¹Ø¿¡ ÀÖ´Â ³ðÀÌ ´Ù¸¥ µÑ º¸´Ù ´õ ¾Æ·¡·Î ³»·Á°¡ ÀÖ½À´Ï´Ù. ±×µµ ±×·² °ÍÀÌ ¾Æ¸¶ limsupÀÇ 'p'¶§¹®¿¡ ±×·± °Í °°½À´Ï´Ù. * '''½Ãµµ II''': ¹®ÀÚÀÇ ³ôÀÌ(height)¿Í ±íÀÌ(depth)°¡ ¼·Î ´Þ¶ó¼, À§ ¾Æ·¡·Î µé¾¦³¯¾¦ÇÏ´ø ·çÆ® ±âÈ£¸¦ \mathstrutÀ» »ç¿ëÇؼ ÇØ°áÇß´ø ±â¾ïÀÌ ÀÖ´ø ÅͶó, ¹Ù·Î À̰ű¸³ª ÇÏ°í ´ÙÀ½°ú °°ÀÌ ½ÃµµÇغýÀ´Ï´Ù. {{{#!vim tex $$ \mathop{\mathstrut\lim}_{n\to\infty}x_n {\rm\ exists} \iff \mathop{\mathstrut\limsup}_{n\to\infty}x_n = \mathop{\mathstrut\liminf}_{n\to\infty}x_n $$ \}}} °á°ú´Â ´ÙÀ½°ú °°¾Ò½À´Ï´Ù. $$\mathop{\mathstrut\lim}_{n\to\infty}x_n {\rm\ exists} \iff \mathop{\mathstrut\limsup}_{n\to\infty}x_n = \mathop{\mathstrut\liminf}_{n\to\infty}x_n$$ È¿°ú°¡ ÀÖ¾ú½À´Ï´Ù. $$n\to\infty$$ ÀÇ À§Ä¡°¡ ÀÏÁ¤ÇØÁ³½À´Ï´Ù. À¢ÀÏ·Î ÀÛÀº³ª¹«°¡ µÎ ¹ø¸¸¿¡ ÇØ°áÇϳª ½Í¾î¼, ÀÚ¼¼È÷ º¸´Ï ¾Æ´Ï³ª ´Ù¸¦±î, $$n\to\infty$$ °¡ ÀÏÁ¤ÇØÁö±ä ÇßÁö¸¸, Á¤´äº¸´Ù ³Ê¹« ¾Æ·¡·Î ³»·Á°¬½À´Ï´Ù. Áï ÇÏÇâ ÆòÁØÈ°¡ µÇ¾îÀÖ¾ú½À´Ï´Ù. ¿©±â±îÁö°¡ ÀÛÀº³ª¹«°¡ ¾Ë°í ÀÖ´Â Áö½ÄÀ¸·Î ÇÒ ¼ö ÀÖ´Â ÀüºÎ¿´½À´Ï´Ù. »ç½Ç À§ÀÇ ¼ö½Äµµ ±×·±´ë·Î º¸±â ÁÁ¾Ò½À´Ï´Ù. ¾î·µç ´äÀº ¾Æ´Ï¹Ç·Î, ÇØ°áÇϱâ À§Çؼ Ã¥À» ºÃ½À´Ï´Ù. ¿ª½Ã ÇØ´äÀº Ã¥¿¡ ÀÖ½À´Ï´Ù. 178ÂÊ¿¡ ´ÙÀ½°ú °°Àº ÇØ°á¹æ¹ýÀÌ Á¦½ÃµÇ¾î ÀÖ¾ú½À´Ï´Ù. \smash¿Í \vphantomÀ» Àß Á¶ÇÕÇؼ »ç¿ëÇϸé, ¿øÇÏ´Â ³ôÀÌ¿Í ±íÀÌ·ÎµÈ ¼ö½ÄÀ» ¸¸µé ¼ö ÀÖ´Ù´Â °ÍÀ̾ú½À´Ï´Ù. ¹Ù·Î ÀÌ°ÍÀÌÁÒ. limsup°ú liminfÀÇ ³ôÀÌ¿Í ±íÀ̸¦ Çϳª·Î ÅëÀÏÇÏ¸é ¹®Á¦°¡ ÇØ°áµÇ´Â °ÍÀ̾ú½À´Ï´Ù. ±×·¡¼ limsupÀÇ ³ôÀÌ¿Í ±íÀ̸¦liminfÀÇ ±×°Íµé¿¡ ¸ÂÃß±â·Î Çß½À´Ï´Ù. liminf¸¦ limsup¿¡ ¸ÂÃâ±îµµ »ý°¢ÇغÃÁö¸¸, ±×·¯¸é lim, liminf µÎ °³¸¦ ¼ÕºÁ¾ß ÇϹǷΠ°üµ×½À´Ï´Ù. :) * '''½Ãµµ III''': ¹æ¹ýÀº ÀÏ´Ü limsupÀÇ ³ôÀÌ¿Í ±íÀ̸¦ ¹«½ÃÇؼ 0À¸·Î ¸¸µì´Ï´Ù. À̶§ »ç¿ëÇÏ´Â ¸í·É¾î°¡ ¹Ù·Î \smashÀÔ´Ï´Ù. ±×¸®°í ³ª¼ liminf¿¡ ¸ÂÃä´Ï´Ù. ±×·¯±â À§Çؼ´Â liminfÀÇ ³ôÀÌ¿Í ±íÀ̸¦ ¾Ë¾Æ¾ß Çϴµ¥, À̶§ »ç¿ëÇÏ´Â ¸í·É¾î°¡ ¹Ù·Î \vphantom{\liminf} ¿´½À´Ï´Ù. ±×·¡¼ ÀÌ µÑÀ» °°ÀÌ »ç¿ëÇϸé, ¾Æ·¡¿Í °°ÀÌ µË´Ï´Ù. {{{#!vim tex \mathop{\smash\limsup\vphantom\liminf} \}}} ±×·¡¼ ÃÖÁ¾ °á°ú´Â ´ÙÀ½°ú °°½À´Ï´Ù. {{{#!vim tex $$ \lim_{n\to\infty}x_n {\rm\ exists} \iff \mathop{\smash\limsup\vphantom\liminf}_{n\to\infty}x_n = \liminf_{n\to\infty}x_n $$ \}}} µåµð¾î ¼¼¹øÀÇ ½Ãµµ ³¡¿¡ ¿øÇÏ´ø °á°ú¸¦ ¾ò¾ú½À´Ï´Ù. ¼¼ ¹ø¸¸¿¡ ¼º°øÇϸé ÀßÇÑ °Í ¾Æ´Õ´Ï±î? :) $$\lim_{n\to\infty}x_n {\rm\ exists} \iff \mathop{\smash\limsup\vphantom\liminf}_{n\to\infty}x_n = \liminf_{n\to\infty}x_n$$
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