cauchy sequence ${a_n}$ in $(0,1)$, ${f(a_n)}$ is also cushy sequence

Question

Let $f(x)$ be a differentiable real valued function on $(0,1)$, and $f'(x)$ is bounded on $(0,1)$, then, I think for every cauchy sequence ${a_n}$ in $(0,1)$, ${f(a_n)}$ is also cushy sequence.

This is because of mean value theorem.

My question is, if $f'(x)$ is not bounded, then, could you tell me the counter example of this statement? That is, cushy sequence which does not converge in $(0,1)$.

Thank you for your help.

Answer 1

Take $f(x)=\frac 1 x$ on $(0,1)$. $(\frac 1n)$ is a Cauchy sequence but $(f(\frac 1n))$ is not Cauchy.