Let $X$ be a locally compact metric space, and $A\subseteq X$ be a bounded closed subset. We want to show that $A$ is complete.
It is well-known that $X$ is compact if and only if it is complete and totally bounded. However, I'm not sure how to approach this problem. Any hints to help me get started would be greatly appreciated.
This is not true. Take $X=(0,1)$ with the usual topology. It is locally compact. But $X$ is a closed and bounded subset of itself and it is not complete.