Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $f:X\to Y$ be a function. Then $f$ is not uniformly continuous if and only if there exist a number $\varepsilon_0>0$ and sequences $(x_n)_{n=1}^{\infty}$ and $(y_n)_{n=1}^{\infty}$ in $X$ such that