We would show that uniform continuous functions preserve Cauchy sequences. That is,
$$ \textrm{If f is a uniformly continuous function on S, and}~s_n~\textrm{is Cauchy sequence on S, then}~(f(s_n))~\textrm{is also a Cauchy Sequence} $$
Here is the proof,
$$\textrm{Since, f is uniformly continuous on S, we have}$$
$$\textrm{for each}~ \epsilon \gt 0, \exists \delta_{\epsilon}, ~\textrm{such that} ~ \forall x, y \in S, ~ |x-y| \lt \delta_{\epsilon} \implies |f(x)-f(y)\lt \epsilon$$
$$since (s_n) ~\textrm{is a Cauchy sequence, we have an N such that} ~ n, m \gt N \implies |s_n - s_m| \lt \delta_{\epsilon}$$
$$\textrm{since} ~s_n, s_m \in S~ ~ |s_n - s_m| \lt \delta_{\epsilon} \implies |f(s_n) - (s_m)| \lt \epsilon$$
$$\forall n,m \gt N~\textrm{we have} |f(s_n) - f(s_m) | \lt \epsilon$$
Is my proof well-written?
P.S. I do not know why the latex code is not rendered.
