Basic Analysis I — Real Analysis · Problem 7.4.19

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · completeness-and-compactness

Problem

Consider two metrics on R\mathbb{R}. Let d(x,y):−∣x−y∣d(x,y) \coloneq |x-y| be the standard metric, and let d′(x,y):−∣x1+∣x∣−y1+∣y∣∣d'(x,y) \coloneq \left|\frac{x}{1+|x|} - \frac{y}{1+|y|}\right|. a) Show that (R,d′)(\mathbb{R}, d') is a metric space (if you have done Exercise 7.3.10, the computation is the same). b) Show that the topology is the same, that is, a set is open in (R,d)(\mathbb{R}, d) if and only if it is open in (R,d′)(\mathbb{R}, d'). c) Show that a set is compact in (R,d)(\mathbb{R}, d) if and only if it is compact in (R,d′)(\mathbb{R}, d'). d) Show that a sequence converges in (R,d)(\mathbb{R}, d) if and only if it converges in (R,d′)(\mathbb{R}, d'). e) Find a sequence of real numbers that is Cauchy in (R,d′)(\mathbb{R}, d') but not Cauchy in (R,d)(\mathbb{R}, d). f) While (R,d)(\mathbb{R}, d) is complete, show that (R,d′)(\mathbb{R}, d') is not complete.

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