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The task of finding a limit point of a sequence is 'simple'; if the sequence converges, we locate the limit point by inspection and proved that it the sequence converges to that point. By uniqueness, there is no other limit point.

For cluster points or accumulation points, since they are not unique, how do I know for sure that I have found them all? Must I prove that there are no other cluster points besides those I have found? Am I being too cautious? What is the importance of cluster points other than the limit point of some subsequence?

Tim
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  • Finding the limit of a sequence is not necessarily simple. Often we prove that the sequence $(a_n)$ has a limit by showing that the sequence is increasing and bounded above. Finding an explicit expression for the limit may be very difficult. – André Nicolas Aug 28 '12 at 02:17

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Yes, in order to be sure that you have found them all, you must show no other cluster points exist. This will most likely be difficult in practice as you need to find a neighborhood of the point which contains no points of the sequence beyond a certain index.

Intuitively, I think about cluster points as the set of points that sequence continues to get near as the index increases. They are also the idea that leads to limit points in topology.

Kris Williams
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  • Coincidentally, a question about the cluster points of the sequence $a_n=\cos n$ was asked today. The answer turns out to be simple, but proving it involves some work. And for closely related sequences the answer is not known. – André Nicolas Aug 28 '12 at 02:30
  • @AndréNicolas: The answer there suggests that any point in [-1,1] is a cluster point. We still need to do some work to prove that points outside [-1,1] is NOT a cluster point. Also, I think we can use the fact that the cluster points form a closed set to check if our answer make sense: suppose you proved that (-1,1) is the set of cluster point, then we know that we have missed the points -1 and 1. By taking closure of the current set of cluster points, we may find out more cluster points of the sequence. – Tim Aug 28 '12 at 02:58
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    It is clear that nothing outside $[-1,1]$ can be a cluster point, since, for example, if $x=1.001$ then no point of the sequence is $\lt 0.001$ from $1.001$. – André Nicolas Aug 28 '12 at 03:10
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A special case:Suppose a sequence {xn=x} is constant then its cluster point is same x

Nadeem
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