Uniform convergence

Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions. The sequence $(f_n)$ converges uniformly to $f$ if for every $\varepsilon > 0$ there is $N$ such that $$|f_n(x) - f(x)| < \varepsilon \quad \text{for all } n \ge N \text{ and all } x \in S.$$

Remarks

Here $N$ depends only on $\varepsilon$, not on $x$. Uniform convergence implies pointwise convergence. Equivalently, $\sup_{x \in S} |f_n(x) - f(x)| \to 0$ as $n \to \infty$. For example, $\frac{\sin(n x)}{n} \to 0$ uniformly on $\mathbb{R}$. Mathlib defines it as TendstoUniformly.

Explanations

L1 human

Uniform convergence means that the whole graph of $f_n$ lies in a thin band around the graph of $f$. The width of the band tends to $0$, at the same rate for all points.