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Prove that: let be $M$ a open and bounded set of $\mathbb{R}^N$. Taking $K = \overline{M}$, then the set $C^k(K)$ of functions $k$ times differentiable is complete with norm $\Vert \cdot \Vert = \sup_{x\in K,|\alpha|\leq k}|D^{\alpha}f(x)|$

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For simplicity, I suggest starting with $N = 1$ and $k = 1$. –  froggie Aug 19 '12 at 20:41
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I would suggest providing some information as to how much you know about the concepts involved, and where precisely you are stuck. –  Ben Millwood Aug 19 '12 at 21:54
    
I need idea to begin, ... –  juaninf Aug 20 '12 at 1:21
    
First, take @froggie's suggestion. Show that a uniformly convergent sequence of continuous functions has a continuous limit. Then it will follow that each $D^\alpha f$ also has a continuous limit. The only tricky part is showing that the limit of the derivatives is the derivative of the limit function. For this you can use Taylor's expansion. –  copper.hat Aug 20 '12 at 3:14
    
for inspiration see this answer –  Norbert Aug 20 '12 at 9:32
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