I was reading this theorem in my analysis text and came across something I could not quite follow:
Theorem: Let S be a subset of the metric space E. Then S is closed if and only if, whenever is a sequence of points of S that is convergent in E, we have .
I understand the "only if" part. Now for the "if" part the text states:
"Suppose S is not closed. Then the complement is not open, and there exists a point p in the complement of S such that any open ball of center p contains points of S. *Hence for each positive integer n we can choose such that .Then , with each and .*This shows that if the hypothesis on convergent sequences holds, then S must be closed, completing the proof. "
^^ I've bolded the text that I don't understand. I've thought about this and I can't figure out where the 1/n comes from as well as the conclusion . Thanks for the help!