Find an example of a complete bounded metric space which is not compact. I think the infinite dimensional sphere satisfies the following criteria.  However, I was hoping that someone could come up with a more elementary example.  Thanks.
Find an example of a complete bounded metric space which is not compact.
 A: Other than discrete spaces, you can take the following general approach. Take any non-compact metric space which is complete. To turn it into a bounded metric space without changing its non-compactness nor its completeness, just change the metric to $\min\{d(-,-),1\}$. So, for instance, $\mathbb R$ is complete, not compact, nor bounded (with the usual metric). After truncating the metric as above you get $\mathbb R$ with a metric such that it is bounded, is complete, but not compact. This little trick shows why in the context of a metric the concept of total boundedness is more useful than boundedness.  
A: Hint: In a Banach space the closed unit ball is compact if and only if the dimension is finite; but a closed subset of a complete metric space is complete.
A: If you want a "simple" example, then I can't think of anything different than a subset of a Banach space. Being complete means being close (since the whole space is complete), and for a closed bounded subset of a vector space to be non-compact, you have to be in an infinite dimensional vector space.
