Every manifold is locally compact? 
Theorem. Every Manifold is locally compact.

This is a problem in Spivak's Differential Geometry.
However, don't know how to prove it. It gives no hints and I don't know if there is so stupidly easy way or it's really complex. 
I good example is the fact that Heine Borel Theorem, I would have no clue on how to prove it if I didn't see the proof. 
So can someone give me hints. I suppose if it's local, then does this imply that it's homeomorphic to some bounded subset of a Euclidean Space?

I do not think the above answers are completely right, since the "Hausdorff" condition in the definition of topological manifolds must be needed.
The key is to prove the following: If $V\subset U\subset X$, and X is Hausdorff, $\bar{V}_{U}$ is compact. Then $\bar{V}_{U}=\bar{V}$.
Proof: By definition, we only need to show that $\bar{V}\subset \bar{V}_U$. Suppose $x\in\bar{V}$, if $x\notin \bar{V}_U$. Since X is Hausdorff and $\bar{V}_{U}$ is compact, $\bar{V}_{U}$ is closed in X. So $ (\bar{V}_{U})^{c}$ is open and $x\in (\bar{V}_{U})^{c}$. Since $x\in \bar{V}$, we must have $V\cap (\bar{V}_{U})^{c}$ is not empty. It is impossible since $\bar{V}_{U}$ contains $V$.
