00. > 0 be given. There is an no (c) such that for all n, m ~ no(c).
Since M is closed, U E M. Thus, M is compact. 0 U Proposition 4. Each relatively compact set is bounded. Proof. Let the set M be relatively compact and suppose that M is not bounded. Then, there exists a sequence (un) in M such that for all n. (30) Since M is relatively compact, there exists a convergent subsequence (un')' Hence (Un') is bounded. This contradicts (30). 0 Example 5. Let M be a subset of JR equipped with the usual norm lui· Ilull := Then, M is relatively compact iff it is bounded.