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.
- "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".
- "If l is the limit of $\{x_n\}$, then every nbd of l contains all but a finite numbers of its members "
Best Answer
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)