Suppose that the sequence converges to a point x, prove that

In addition, when would x be an accumlation point.

Proof so far.

So since , , there exist such that whenever , we would have .

Now, I need to show that this x is in the closure, which is the intersection of all closed set containing .

So my idea is, for any closed set that contains the sequence, suppose that x is not in them, then x is not in the set that the sequence is in, then but the completeness property, that is impossible.

And x would be an accumlation point if the set is closed.

Is this right?