Basic Analysis I — Real Analysis · Problem 7.1.9

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

Let (X,d)(X, d) be a nonempty metric space and S⊂XS \subset X a subset. Prove: a) SS is bounded if and only if for every p∈Xp \in X, there exists a B>0B > 0 such that d(p,x)≤Bd(p,x) \leq B for all x∈Sx \in S. b) A nonempty SS is bounded if and only if diam⁡(S):−sup⁡{d(x,y):x,y∈S}<∞\operatorname{diam}(S) \coloneq \sup\{d(x,y) : x, y \in S\} < \infty.

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