Uniform continuity preserves Cauchy sequences

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:X\to Y$ be uniformly continuous.

Proposition. If $(x_n)$ is a Cauchy sequence in $X$, then $(f(x_n))$ is a Cauchy sequence in $Y$.

Indeed, given $\varepsilon>0$, choose $\delta>0$ from uniform continuity. For all sufficiently large $m,n$, the Cauchy property gives $d_X(x_m,x_n)<\delta$, hence $d_Y(f(x_m),f(x_n))<\varepsilon$.

Continuity alone does not suffice: $f(x)=1/x$ on $(0,1)$ sends the Cauchy sequence $x_n=1/n$ to the non-Cauchy sequence $n$.