Lebl-Basic-Analysis-I · Problem 7.3.2

Metric Spaces

Mathematics · sequences-and-convergence

Problem

a) Show that d(x,y)≔min⁡{1,∣x−y∣}d(x, y) \colonequals \min\{1, |x - y|\} defines a metric on R\mathbb{R}. b) Show that a sequence converges in (R,d)(\mathbb{R}, d) if and only if it converges in the standard metric. c) Find a bounded sequence in (R,d)(\mathbb{R}, d) that contains no convergent subsequence.

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