Proposition: Uniform Convergence Criterion of Cauchy

Let $\mathbb F$ be a either the field of real numbers or the field of complex numbers and let $D\subset \mathbb F.$ The sequence of functions $f_n:D\to\mathbb F$ is uniformly convergent to a function $f:D\to\mathbb F$ if and only for every $\epsilon > 0$ there is an index $N$ such that the supremum norm $||f_n-f_m||_\infty < \epsilon$ for all $n,m\ge N.$

Proofs: 1

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition