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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.

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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.

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Unfortunately, there doesn't seem to be one standard definition for theses terms. One author may say that accumulation points are limit points, others do not. Best thing to do is to check the definition given in book, paper, etc that you are reading.

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  • $\begingroup$ So it isn't a big deal to treat them somewhat interchangeably? $\endgroup$ – Aaron Zolotor Feb 2 '16 at 15:14
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    $\begingroup$ It might depending on how these terms are defined in whatever source you're reading. $\endgroup$ – Tim Raczkowski Feb 2 '16 at 15:16

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