uniform limit

Def pg. 243

A sequence of function $(f_n)$ on $A\subseteq \mathbb{R}$ to $\mathbb{R}$ converges uniformly on $A_0\subseteq A$ to a function $f:A_0⟶\mathbb{R}$ if for all $x\in A_0$
$(\forall \epsilon >0)(\exists K(\epsilon ))(\forall n\ge K(\epsilon ))|f_n(x)-f(x)|<\epsilon$

Claim 18.2

${\lim }_{n⟶\infty }{d}_{\infty }(f_n,f)=0$