Cauchy's Uniform Convergence Criterion

Definition

Suppose $f_n:\mathcal{D}\to \mathbb{R}$ If $\forall ϵ>0,\exists N\in \mathbb{N},(\forall x\in \mathcal{D},\forall m,n\in \mathbb{N},m>n\ge N\Rightarrow \left|\sum _{k=n+1}^m{f}_k(x)\right|<ϵ)$, then $\sum _{n=1}^{\infty }{f}_n(x)$ is uniformly convergent on $\mathcal{D}$