(a) Show that n d(m, n) |m2 – n?] 1 + \m2 – n2] defines a metric on the set of natural numbers N. Hint: use the monotoni

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A Show That N D M N M2 N 1 M2 N2 Defines A Metric On The Set Of Natural Numbers N Hint Use The Monotoni 1 (84.55 KiB) Viewed 21 times

(a) Show that n d(m, n) |m2 – n?] 1 + \m2 – n2] defines a metric on the set of natural numbers N. Hint: use the monotonicity of the function t t/(1 + t) for t E [0, 0o). (b) Does d define a metric on the set of integers Z? Justify your answer. (c) Describe all bounded subsets of (N, d). (d) Determine the largest rı > 0 and the smallest r2 > 0 such that, for all me N, B°(m, rı) = {m} and Bºm, r2) = N. = (e) Describe all totally bounded subsets of (N, d).