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?