(a) Show that d(m, n) = |m2 - n?! 1 + 1m2 - 121 defines a metric on the set of natural numbers N. Hint: use the monotoni

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A Show That D M N M2 N 1 1m2 121 Defines A Metric On The Set Of Natural Numbers N Hint Use The Monotoni 1 (31.69 KiB) Viewed 18 times

(a) Show that d(m, n) = |m2 - n?! 1 + 1m2 - 121 defines a metric on the set of natural numbers N. Hint: use the monotonicity of the function to t/(1+1) for t€ [0,00). (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 ri> 0 and the smallest r2 > 0 such that, for all me N, Bºm, n) = {m} and Bºm, r2) = N. (e) Describe all totally bounded subsets of (N, d).