# $k$ times differentiable functions are complete

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
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