TITLE: Difference between a limit and accumulation point?
QUESTION [14 upvotes]: What is the exact difference between a limit point and an accumulation point?
An accumulation point of a set is a point, every neighborhood of which has infinitely many points of the set. Alternatively, it has a sequence of DISTINCT terms converging to it?
Whereas a limit point simply has a sequence which converges to it? i.e. something like $(1)^n$ which is a constant sequence.
Is this the right idea? As much detail and intuition as possible would be greatly appreciated.

REPLY [8 votes]: The difference is very simple.
1) As you wrote: an accumulation point of a set is a point, every neighborhood of which contains infinitely many points of the set. 
2) But a limit point is a special accumulation point. No matter how small neighborhood you choose, all members $a_n$ (after a certain $n$) are in the neighborhood of the limit point.  
The requirement for all members (after certain $n$) is obviously stronger than the requirement for infinitely many points/members.     
So every limit point is an accumulation point, but not every accumulation point is a limit point. Also note that: (1) if a sequence has a limit point, then that's the only accumulation point of the sequence; (2) if a sequence has more than one accumulation points, that this sequence has no limit point. Try to prove these two, it will clear your confusions.