Question:
Convergent subsequences
1. If A and U are two subsets of a normed vector space, and U is open, show that A+U is open. Here A+U={a+u | a belongs to A and u belongs to U}.
2. Suppose {xn} from n=p to infinity is a non-converging subsequence in a compact set of a metric space. Show that it has two convergent subsequences which converge to distinct limit points.