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Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions that are uniformly continious on a compact set $K$ and that converges pointwise to a function $f$ that is uniformly continious on $K$, too. Can I know from this that the sequence converges uniformly to $f$?

I am not sure about that.

To my knowledge if a sequence of functions is continious and it converges uniformly to a function, then this functions is continious, too. But if both the sequence and the limit function are uniformly continious I do not know if I can then say that the convergence is uniformly.

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    $\begingroup$ No. On $[0,1]$, let $f_n$ be the function taking the value $1$ at $1/n$, $0$ at $\{0\}\cup[2/n,1]$ and piecewise linear where it is not yet defined. (Incidentally, continuous on a compact set implies uniform continuity.) What you want is equicontinuity. Google "Arzela-Ascoli theorem". $\endgroup$ – David Mitra May 29 '14 at 10:26
  • $\begingroup$ Hm, sorry, do not understand Arzela Ascoli at all. ;( $\endgroup$ – mathfemi May 29 '14 at 11:16

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