# Definition(s) > [!NOTE] Definition 1 (Uniform convergence of sequence of real functions) > Let $f_{n}:\Omega\to \mathbb{R}$ be a [[Sequences|sequence]] of [[Real Function|real functions]]. We say that $(f_{n})$ converges uniformly to $f:\Omega\to \mathbb{R},$ denoted $f_{n}\rightrightarrows f,$ iff for all $\varepsilon>0,$ there exists $N$ such that for all $n>N,$ $\forall x\in \Omega: \quad |f_{n}(x)-f(x)|<\varepsilon.$ > [!NOTE] Definition 2 (Uniform convergence of sequence of real functions in terms of supremum norm) > Equivalently, $f_{n}\rightrightarrows f \iff \forall\varepsilon>0, \exists N\in \mathbb{N} \text{ such that } \forall n>N, \; \lvert \lvert f_{n}-f \rvert \rvert_{\infty} <\varepsilon $where $\lvert \lvert f \rvert \rvert_{\infty}=\sup_{x\in \Omega}|f(x)|,$ denotes the [[Supremum (Uniform) Norm|supremum norm]]. > [!NOTE] Definition 3 (Uniform convergence of sequence of real functions in terms of supremum norm) > Equivalently, $f_{n}\rightrightarrows f \iff \lim_{ n \to \infty } ( \lvert \lvert f_{n}-f \rvert \rvert_{\infty}) =0 $where $\lim_{ n \to \infty }$ denotes [[Convergence|limit of real sequence]]. > [!Example] Example > Consider for example $f_{n}(x)= \frac{\sin(n^2x)}{n}.$ Then using the [[Sandwich Rule|sandwich rule]], $f_{n}\rightrightarrows0$ since $\forall x: \ \left\lvert \frac{\sin(n^2x)}{n} \right\rvert \leq \frac{1}{n} \to 0, \quad \text{as } n\to \infty $ # Properties(s) # Application(s) **More examples**: # Bibliography