Basic Analysis I — Real Analysis · Problem 7.1.10

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

a) Working in R\mathbb{R}, compute diam⁡([a,b])\operatorname{diam}([a,b]). b) Working in Rn\mathbb{R}^n, for every r>0r > 0, let Br:−{x12+x22+⋯+xn2<r2}B_r \coloneq \{x_1^2 + x_2^2 + \cdots + x_n^2 < r^2\}. Compute diam⁡(Br)\operatorname{diam}(B_r). c) Suppose (X,d)(X, d) is a metric space with at least two points, dd is the discrete metric, and p∈Xp \in X. Compute diam⁡({p})\operatorname{diam}(\{p\}) and diam⁡(X)\operatorname{diam}(X), then conclude that (X,d)(X, d) is bounded.

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