TITLE: Confusion on the definition of accumulation points
QUESTION [5 upvotes]: I've been trying to learn a bit about limits of sequences and accumulation points to get a better intuition behind the workings for calculus and I got confused on the definitions of limits,limit points and accumulation points of sequences and sets.
My first question is a limit of a sequence the same as the accumulation point and is that the same as the limit point i looked online and its all very vague.
My second confusion is that is the limit of a sequence the same as the limit of a set if not is there some proof or intuitive explanation as to why not?.
I know this is probably a very simple and probably trivial concept for all of you on here but its confused me a lot.
Thanks in advance

REPLY [2 votes]: There is a difference between limit and limit point.
The concept is defined for sequences and functions but limit point is defined for sets, as mentioned in above answer. A sequence may have limit point but no limit. For example let $\{a_n\}$ is defined as $$1+\frac{1}{n} , (-1)+ \frac{1}{n},... $$
That $a_n=1+\frac{1}{n} $ for odd n's and $a_n=-1+\frac{1}{n} $ for evens.
In this sequence both $1$ and $-1$ are limit point but the sequence is not convergent and there is no limit.