Hi guys.

What are the ways to determine if a sequence of functions $\displaystyle f_n(x)$ uniformly converges to a function $\displaystyle f(x)$?

I know how to do it using the definition of uniform convergence, but sometimes it is not so easy to prove it using the definition.

The lecturer in class mentioned something about examing the supremum of $\displaystyle \left \{ f_n(x) \right \}$ in the domain, and then finding $\displaystyle \lim_{n\rightarrow \infty }\sup$, but I couldn't understand what does that have to do with it.

for example:

$\displaystyle f_n(x)=\frac{\arctan x}{n}$ in $\displaystyle \mathbb{R}$.

it easy to show that this sequence converges to $\displaystyle f(x)=0$, it is also easy to prove, using the definition, that it is uniformly converges to $\displaystyle f(x)$, but this is how he proved it:

"in order to show that $\displaystyle f_n(x)$ uniformly converges to $\displaystyle f(x)$, we'll use '$\displaystyle \lim\_\sup$ test' (as he called it):

$\displaystyle \sup_{x\in\mathbb{R}}\left \{ \frac{\arctan x}{n} \right \}=\frac{\pi}{2n}$

and then:

$\displaystyle \lim_{n\rightarrow \infty }\frac{\pi}{2n}=0$

and therefore $\displaystyle f_n(x)$ uniformly converges to $\displaystyle f(x)$."

my question is: why?

and... are there other methods to prove uniform convergence of functions?

thanks in advanced!