※ 引述《PttFund (批踢踢基金)》之銘言：
: Let f: (S,d_S) ---> (T,d_T) is uniformly continuous on S and
: {x_n} is a Cauchy sequence on S. Show that {f(x_n)} is a Cauchy
: sequence in T.
proof : Since f is uniformly continuous,
given ε > 0 , there exists δ > 0 such that d(x,y) < δ
=> d(f(x) , f(y)) < ε
Since {x_n} is a Cauchy sequence,
for δ > 0 , there exists N such that n ≧ N , d(x_n , x_m) < δ
=> d(f(x_n) , f(x_m)) < ε
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