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I am reading the limit point of sequence and limit of sequence. I can't understand the difference between limit point of sequence and limit of sequence. In my book they told me following two points.

  1. "If l is the limit point of $\{x_n\}$ , then every nbd of l containing an infinite number of its members does not exclude the possibility of an infinite number of members of $\{x_n\}$ lying outside that nbd".
  2. "If l is the limit of $\{x_n\}$, then every nbd of l contains all but a finite numbers of its members "
user354069
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    Example: $1$ is a limit point of the sequence $(x_n)$ defined by $x_n=(-1)^n$, but not its limit. (And one should say "a" limit point, not "the" limit point.) – Did Sep 18 '16 at 17:09
  • What you call "a limit point" is usually called "accumulation point" See eg http://math.stackexchange.com/questions/374859/ and http://math.stackexchange.com/a/1637493/312 – leonbloy Sep 18 '16 at 17:12
  • which book were you reading? – blabla Jul 17 '18 at 12:25

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limit point and limit of a sequence are two different concepts, $1$. consider the sequence $(-1)^n=(-1,1-1,1,-1,\cdots)$ Here $-1$ and $1$ are the two limit points of this sequence first take 1 then every open interval containing 1 contains all even numbered terms i.e $(1,1,1,1,\cdots)$ similarly for $-1$. Note:(if L is a limit point of a sequence (an) then actually this L is limit of a subsequence of (an)

Harsh Kumar
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Muhammad Tahir
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