Find an example of a complete metric space $X$ in which all sufficiently small closed balls are compact but large ones are not.
First of all, what is ''a small ball'' and ''a large ball'' in a metric space? Does it have to do with the radius of the ball?
Theorem: Every closed ball of $X$ is compact $\Leftrightarrow$ $X$ has the nearest-point property.
Since $X$ has not the nearest-point property, it is not compact. But $X$ is complete so the space is not total bounded.
I think this is the fact that we want to take advantage of, to prove that large balls are not compact.
Not sure how to proceed or if this thinking is correct.
The nearest-point property is not in many topology books, so i give the definition:
$(X,d)$ is a metric space. The following are equivalent:
i) Every infinite bounded subset of $X$ has an accumulation point in X(BW criterion)
ii) Every bounded sequence of $X$ has a convergence subsequence in $X$.
iii) X is complete and every bounded subset of $X$ is totally bounded.