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

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A Show That M2 N2 D M N 1 M2 N2 Defines A Metric On The Set Of Natural Numbers N Hint Use The Monotonicit 1 (58.45 KiB) Viewed 19 times

(a) Show that |m2 – n2| d(m,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 € (0,0). (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 m EN, B° (m,rı) = {m} and Bºm, r2) = N. (e) Describe all totally bounded subsets of (N,d).