Let $E$ be a set, let $(Y,d_Y)$ be a [complete metric space](/page/Complete%20Metric%20Space), and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n:E \to Y$. Then the following are equivalent:
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1. The sequence $(f_n)_{n=1}^{\infty}$ is uniformly Cauchy on $E$, meaning that for every $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that, for all $m,n \ge N$ and all $x \in E$,
2. There exists a function $f:E \to Y$ such that $(f_n)_{n=1}^{\infty}$ converges uniformly to $f$ on $E$, meaning that for every $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that, for all $n \ge N$ and all $x \in E$,