Real Analysis mathematics (please help)

Question

Let f:(0,1)→R be uniformly continous. Let (x_n) where n goes from 1 to infinity, be a sequence in (0,1). Suppose that lim_n→infinityx_n= 0 . Prove that lim_n→infinityf(x_n) exists

Answer 1

Since x_n → 0, then {x_n} is a Cauchy sequence in (0,1). By uniform continuity of f, the sequence {f(x_n)} is Cauchy in R. Completeness of R implies lim f(x_n) exists.