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scottyg88
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Homework Statement
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.
Homework Equations
From "introduction to Analysis" Rosenlicht, page 47.
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 :)