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## Main Question or Discussion Point

Theorem: Let S be a compact subset of ℝ^n. Then S is closed.

Before looking at the book I wanted to come up with my own solution so here is what I've thought so far:

Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a finite subcover of that open covering and it consists of neighborhoods. Now pick the neighborhood with maximum radius n' and consider the neighbourhood A with radius n'+1. Clearly A is open and S is a subset of A and additionally, A\S is nonempty and still open(?) (this part really needs verification). Hence, ℝ\(A\S)=ℝ\A disjoint union S is closed. Does that mean that S is closed? If not, why?

Before looking at the book I wanted to come up with my own solution so here is what I've thought so far:

Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a finite subcover of that open covering and it consists of neighborhoods. Now pick the neighborhood with maximum radius n' and consider the neighbourhood A with radius n'+1. Clearly A is open and S is a subset of A and additionally, A\S is nonempty and still open(?) (this part really needs verification). Hence, ℝ\(A\S)=ℝ\A disjoint union S is closed. Does that mean that S is closed? If not, why?