$f$ uniformly continuous , $a_n$ Cauchy $\Rightarrow f(a_n)$ is Cauchy

Question

Let $I$ be an interval and let $f: I\to \mathbb{ R}$ be uniformly continuous on I. Suppose that $\{a_n\}$ is a Cauchy sequence in $I$. Prove that $\{f(a_n)\} $is a Cauchy sequence.

Answer 1

Fix $\epsilon>0$, Since $f$ is uniformly continuous,$\exists \delta>0 \ni |f(x)-f(y)|<\epsilon$ Whenever $|x-y|<\delta$.

$a_n$ is Cauchy, so $\exists N\in\mathbb{N}\ni |a_n-a_m|<\delta \forall m,n>N$

So $|f(a_n)-f(a_m)|<\epsilon \forall m,n>N$. So $\{f(a_n)\}$ is Cauchy