If $\{a_n\}$ is a Cauchy sequence, and $f$ is continuous, is $\{f(a_n)\}$ a Cauchy sequence?

Question

If $\{a_n\}$ is a Cauchy sequence, and $f$ is continuous, is $\{f(a_n)\}$ a Cauchy sequence?

It's easy to do it on some certain domain of $f$. But the problem is How to disprove it is $f$ is $\Bbb R$ to $\Bbb R$

Answer 1

From $\Bbb R$ to $\Bbb R$, it is true.

Since $\{a_n\}$ is Cauchy in $\Bbb R$ implies $\{a_n\}$ converges and hence by continuity of $f$ , $f(a_n)$ is also converges and hence Cauchy!