Let $A$ and $B$ be two non-empty closed convex sets in Hilbert space $H$. Then the distance between $A$ and $B$ ( i.e.$d(A,B)$ ) is attained when either (a) one set is bounded; or (b) both of them are unbounded, but $d(x,y)\rightarrow \infty$ whenever $\parallel x \parallel \rightarrow \infty$ and $\parallel y \parallel \rightarrow \infty$.
I have known one way to deal with this:
Let $X = B - A$ be the Minkowski sum. Set $v = P_{\bar{X}}(0)$, then $\parallel v \parallel = d(A,B)$, and the distance is attained if and only if $v \in X$. In particularly, it is attained whenever $X$ is closed.
I know that $B-A$ is closed if one set is compact. But I have no idea how to prove $B-A$ is closed under the condition (a) or (b). And my question is: does this way work? If so, how can I prove the rest part?