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Math 115, Homework5Solutions
Problem 4(a) Suppose thatfis not bounded onS. Then there is some sequence(sn)∈Ssuch that|f(sn)| →∞. Now sinceSis bounded, it follows that(sn)is a bounded sequence, so it has a convergingsubsequence(snk).Thensnkis Cauchy, and sincefis uniformly continuous,f(snk)is alsoCauchy, so it converges, which contradicts the fact that|f(sn)| → ∞. Hencefis a boundedfunction.(b) Suppose that1/x2is uniformly continuous on(0,1), Then it would follow thatfis a boundedfunction. This is not true, since