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?
Best Answer
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.