$(x_n)_n$ is a bounded sequence in $(a,b)$

Question

If $(x_n)_n$ is a bounded sequence in $(a,b)$ and $f (x) \colon (a,b) \to \mathbb{R}$ is a uniformly continuous function then show that $(f (x_n))_n$ has a convergent subsequence.

I know that the above holds true if $(x_n)_n$ is Cauchy but if only it is bounded then is it true?

Answer 1

$(x_n)$ is bounded, hence $(x_n)$ contains a convergent subsequence $(x_{n_k})$.

$(x_{n_k})$ is Cauchy, hence $(f(x_{n_k}))$ contains a convergent subsequence, which is a subsequence of $(f(x_n))$.