Prove that if $\displaystyle f:A \to R $ is uniformly continuous on A and $\displaystyle x_n $ is a cauchy sequence in A then $\displaystyle f(x_n) $ is a cauchy sequence in R .

R=real numbers

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- Nov 3rd 2009, 11:47 AMflower3cauchy sequence
Prove that if $\displaystyle f:A \to R $ is uniformly continuous on A and $\displaystyle x_n $ is a cauchy sequence in A then $\displaystyle f(x_n) $ is a cauchy sequence in R .

R=real numbers - Nov 3rd 2009, 06:06 PMredsoxfan325
Fix $\displaystyle \epsilon>0$. Then by uniform continuity, $\displaystyle \exists\delta>0$ s.t. $\displaystyle |x_n-x_m|<\delta \implies |f(x_n)-f(x_m)|<\epsilon$.

Since $\displaystyle \{x_n\}$ is Cauchy, $\displaystyle \forall\delta>0$, $\displaystyle \exists N$ s.t. $\displaystyle n,m>N \implies |x_n-x_m|<\delta$

So...