Suppose that the sequence is in .

Prove that converges if and only if it has a unique cluster point.

My proof:

Given , find the cluster point such that there exist infinite values n with

I know that I need to find a such that whenever , we have .

But which N do I pick that so I can be sure that all the n afterward would get the sequence going into x and not going somewhere else? (In which I assume since the sequence only has one cluster point, the other point cannot go anywhere but x)

For the converse, let , find such that whenever , we have .

Then there exist infinitely values of n such that , which means x is a cluster point.

Now, suppose to the contrary that there exist another point such that there exist infinitely values of n with .

I want to find an N such that the sequence would be converging both to x and y, which is impossible, but how should I go about choosing it?

Thank you!