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 $Y$.

This is an important structural feature: uniform continuity is exactly the hypothesis needed to transport completeness properties through a map.