For $f:X\rightarrow Y,$ if $f$ is continuous and $X$ is a complete metric space, does $f$ preserve Cauchy sequences?

Question

For $f:(X,d)\rightarrow (Y,ρ),$ if $f$ is continuous and $(X,d)$ is a complete metric space, does $f$ preserve Cauchy sequences (i.e. $(x_n)$ is Cauchy in $X$ $\Rightarrow$ $(f(x_n))$ is Cauchy in $Y$)?

If that is true, how to prove this? Is that is not true, may I please ask for a counter-example?

Answer 1

If $(x_n)$ is Cauchy in $X$ then $x_n\to x$ for some $x\in X$ by completeness. Thus, the sequence $(f(x_n))$ satisfies $f(x_n)\to f(x)$ by continuity. Convergent sequences are Cauchy, hence the claim.