(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is the Theorem as stated in the book:

Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have:

lim(n->inf)pn is in S.

2. Relevant equations

From "introduction to Analysis" Rosenlicht, page 47.

3. The attempt at a solution

I understand the "only if" portion of this theorem, in that a closed subset implies the limit will lie in the subset. However, I'm missing something in the "if" portion, in that if a subset contains a convergent sequence and the limit is contained in the subset, then the subset must be closed. Maybe I am reading this wrong, but could it not be the case that a convergent sequence (and its limit) lie completely in an open subset. For example, the sequence 1/n^2 is contained completely in (-1,1), an open subset of the metric space R.

BTW... I've used the info in this forum for a long time... glad to finally be a part of it :)

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# Homework Help: Convergence of Sequences and closed sets

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