Theorem: Let S be a subset of the metric space E. Then S is closed iff whenever p1,p2,p3,... is a sequence of points of S that is convergent in E, we have lim n→∞ p_n ∈ S.
The Attempt at a Solution
I am having trouble understand the "if" part of the proof. It goes:
Suppose S ⊂ E is not closed. Then S^c is not open and there exists a point p ∈ S^c such that any open ball of center p contains points of S. Hence for each positive integer n we can choose p_n ∈ S such that d(p,p_n) < 1/n. Then lim n→∞ p_n = p with p_n ∈ S and p not in S.
Why would there be an open ball of center p containing points of S? Also, why did the book choose 1/n for (what is d(p,p_n) < 1/n telling us)?