real analysis – the Cauchy function sequence converges uniformly

Either a suite of Cauchy functions $ {f_n (x) } $, it is obvious that it converges punctually towards a function $ f (x) $. To prove uniform convergence, the standard argument is:

Let $ varepsilon> $ 0. To choose $ N $ as for all $ m, n geq N $, $ | f_n (x) -f_m (x) | < varepsilon $. To repair $ n $ and takes $ m to infty $. Since $ f_m (x) to f (x) $it follows that $ | f_n (x) -f (x) | leq varepsilon $ for everyone $ n geq N $ and $ x in E $.

I can not find the "take the limit of $ m $ to infinity "quite rigorous because $ f_m (x) to f (x) $ is only by point and depends on $ x $. How can we draw a conclusion from for all $ x in E $ then? What do I forget here?