Uniformly Cauchy continuous functions on $[0,1]$ converge to a continuous limit.Suggest a slogan
Let \(C([0,1])\) be the space of continuous functions \(f\colon[0,1]\to\RR\) with the norm \[ \norm{f}_\infty \coloneqq \sup_{x\in [0, 1]} \abs{f(x)} . \] Then \(C([0,1])\) with the metric \(d(f,g)\coloneqq\norm{f-g}_\infty\) is a complete metric space.