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This was left as an unproved theorem in our class:

Theroem: If $X$ is a finite dimensional normed vector space then each subset $M$ of $X$ is compact if and only if $M$ is closed and bounded.

How do I prove it? I know that $M$ is also finte dimensional and hence complete. So it is bounded.

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    Ok. I've only seen Heine Borel for $\mathbb{R}$. The other fact has already been proven in class.2012-04-10

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$Hint$ If $X$ has dimension $n$ then $X$ is linearly homeomorphic to $\mathbb R^n$.