Sequential criterion for uniform continuity
= Sequential criterion for uniform continuity
A function between metric spaces is uniformly continuous exactly when $d(x_n,y_n)\to0$ implies $d(f(x_n),f(y_n))\to0$ for every two sequences $(x_n)$ and $(y_n)$. Failure of uniform continuity supplies pairs with input distance below $1/n$ but output distance bounded below.