I'm trying to prove that if $\{x_n\}$ is Cauchy sequence which located in $E$ ($f$ is uniformly continuous) then $\{f(x_n)\}$ is a Cauchy.
let say {$x_n$} is a Cauchy sequence in $E$ and $E \to R$ and $f$ is uniformly continuous
since {$x_n$} is Cauchy sequence so there should be exist
$|p_n - p_m| < \epsilon$
and also its uniformly continuous on $E$, this imply that
$|f(p_n)- f(p_m)| $ also has the $\delta$ such as $|f(pn)- f(pm)| < \delta$
then this can attribute to say that
{${f(x_n)}$} is the Cauchy sequence
please take a look my proof and correct it