Let $f$ be a uniformly continuous function on A of $\Bbb{R}$. How do I show that if $a_n$ is Cauchy, then $f(a_n)$ is Cauchy.
This is what I have worked on, but it does not quite make sense since I feel like I didn't really use the given condition that $f$ is uniformly continuous.
Let $\epsilon>0$, $f$ is uniformly continuous, so there exists$\delta>0$ st $|f(x)-f(y)|<\epsilon$ for $|x-y|<\delta$.
Since $a_n$ is Cauchy, there exists $N>0$ such that $|a_n-a_m|<\delta$ for $m,n>N$
Hence$|f(a_n)-f(a_m)|<\epsilon$ for $m,n>N$. So $f(a_n)$ is Cauchy