Let $(X,d)$ be a metric space. If a sequence $(f_n)\subset C_E(X)$ converges to $f\in C_E(X)$ uniformly on $X$, then the family $\mathscr F=\{f_1,\dotsc,f_n,\dotsc,f\}$ is equicontinuous at each $x\in X$.

*Proof.*
Let $\epsilon>0$ and fix $x\in X$. There is a $\delta_x(\epsilon)>0$ such that $||f(y)-f(x)||<\epsilon/3$ whenever $d(x,y)<\delta_x$ since $f$ is continuous at $x$. Also, since $(f_n)$ converges uniformly, there is an $N(\epsilon)\in\Bbb N$ such that $||f_n(y)-f(y)||<\epsilon/3$ whenever $n\ge N$, for any $y\in X$. Under these conditions,