Let f monotone function in $\mathbb{R}$. Then $x_n$ Cauchy implies $f(x_n)$ Cauchy sequence [closed]

I have to prove is true the following, given $f:\mathbb{R}\to\mathbb{R}$, f monotone in $\mathbb{R}$:

$\forall x_n$ sequence in $\mathbb{R}$, if $(x_n)$ is a Cauchy sequence, then $f(x_n)$ is a Cauchy sequence.

I know that if $f$ is uniformly continuous or continous the statement is true.