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.

Proof sketch: Let $\varepsilon>0$. Uniform continuity gives $\delta>0$ such that $d_X(x,y)<\delta$ implies $d_Y(f(x),f(y))<\varepsilon$. Since $(x_n)$ is Cauchy, choose $N$ with $d_X(x_n,x_m)<\delta$ for all $m,n\ge N$. Then $d_Y(f(x_n),f(x_m))<\varepsilon$ for all $m,n\ge N$, so $(f(x_n))$ is Cauchy.