Let $(X, d_X)$ and $(Y, d_Y)$ be [metric spaces](/page/Metric%20Space), and let $f: X \to Y$ be uniformly [continuous](/page/Continuity). If $(x_n)_{n=1}^\infty$ is a [Cauchy sequence](/page/Cauchy%20Sequence) in $X$, then $(f(x_n))_{n=1}^\infty$ is a Cauchy sequence in $Y$.