I want to show that when A is a subset of R^n with the standard topology, and if

Question

I want to show that when A is a subset of R^n with the standard topology, and if p is a limit point of A, then here exist a sequence of points in A that converges to p.

This is what I have:

I know that if p is a limit point of A, so every neighborhood N of p intersects A in a point different than p.

I guess I should start with letting p be a limit point of A, and therefore every neighborhood of p intersects A in a point different than p, say q. But I don't know how to get the sequence part...

I need help! Thank you.

Explanation / Answer

Start with V1(p), let q1 be any element of the intersection between V1(p) and A (you know that q1 must exist because p is a limt point). Now let q2 be any element of the intersection of V1/2(p) and A. Carry on with that pattern, so that qn is any element of the intersection of V1/n(p) and A.

These q's give you a sequence in A that converges to p.

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