# 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?