Uniform Convergence

Definition

$\underset{n\to \infty }{\lim }{f}_n=f\text{ uniformly}$ Suppose $f:\mathcal{D}\to \mathbb{R}$, is a function, and $(f_n)$ is a Sequence of Functions whose term has the same domain as the function $f$ If $\forall ϵ>0,\exists N\in \mathbb{N},\forall x\in \mathcal{D},\forall n\ge N,|f_n(x)-f(x)|<ϵ$, then $(f_n)$ converges uniformly to $f$ on $\mathcal{D}$

Facts

Uniformly convergent Sequence of Functions $(f_n)$ implies Pointwise Convergence