Uniform convergence (sequence of functions)

Let $X$ be a set and let $(Y,d_Y)$ be a metric space. A sequence of functions $f_n:X\to Y$ converges uniformly to $f:X\to Y$ if

\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n\ge N,\ \forall x\in X,\ d_Y\bigl(f_n(x),f(x)\bigr)<\varepsilon.

Equivalently,

\sup_{x\in X} d_Y\bigl(f_n(x),f(x)\bigr)\xrightarrow[n\to\infty]{}0,

where the supremum is taken in $[0,\infty]$ .

Uniform convergence is strong enough to pass many properties to the limit (e.g., continuity, under standard hypotheses). Compare with pointwise convergence. It is a central tool in analysis and approximation theory.

Examples

$f_n(x)=\frac{1}{n}\sin x$ converges uniformly to $0$ on $\mathbb{R}$ .
On $[0,1]$ , $f_n(x)=x^n$ converges pointwise to $f$ (as above) but not uniformly (since $\sup_{x\in[0,1]}|x^n-f(x)|=1$ for all $n$ ).
If $f_n$ are continuous on a compact set and converge uniformly, then the limit is continuous (uniform limit theorem).