# cauchy sequence

• November 3rd 2009, 12:47 PM
flower3
cauchy sequence
Prove that if $f:A \to R$ is uniformly continuous on A and $x_n$ is a cauchy sequence in A then $f(x_n)$ is a cauchy sequence in R .
R=real numbers
• November 3rd 2009, 07:06 PM
redsoxfan325
Quote:

Originally Posted by flower3
Prove that if $f:A \to R$ is uniformly continuous on A and $x_n$ is a cauchy sequence in A then $f(x_n)$ is a cauchy sequence in R .
R=real numbers

Fix $\epsilon>0$. Then by uniform continuity, $\exists\delta>0$ s.t. $|x_n-x_m|<\delta \implies |f(x_n)-f(x_m)|<\epsilon$.

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

So...