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Math Help - how to determine uniform convergence?

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    how to determine uniform convergence?

    Hi guys.
    What are the ways to determine if a sequence of functions f_n(x) uniformly converges to a function 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 \left \{ f_n(x) \right \} in the domain, and then finding \lim_{n\rightarrow \infty }\sup, but I couldn't understand what does that have to do with it.

    for example:

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

    it easy to show that this sequence converges to f(x)=0, it is also easy to prove, using the definition, that it is uniformly converges to f(x), but this is how he proved it:
    "in order to show that f_n(x) uniformly converges to f(x), we'll use ' \lim\_\sup test' (as he called it):

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

    and then:

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

    and therefore f_n(x) uniformly converges to f(x)."
    my question is: why?

    and... are there other methods to prove uniform convergence of functions?
    thanks in advanced!
    Last edited by Stormey; July 13th 2013 at 03:00 AM.
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