I think I have this problem almost done. I am taking $C([0,1])$ to be the set of all continuous function $f\colon[0,1] \to \mathbb{R}$. I have already shown that $\displaystyle\|f\| = \max_{x\in [0,1]} |f(x)|$ is indeed a norm on $C([0,1])$ and makes it a normed space.
My next step was to show that if $f_n$ is a Cauchy sequence in $C([0,1])$ then for all $x \in [0,1]$ $f_n (x)$ is a Cauchy sequence in $\mathbb{R}$. That wasn't too hard.
Now I'm stuck trying to show the pointwise convergence of each sequence, that is $f(x) = \lim_{n \to \infty} f_n (x)$, by showing that $\displaystyle\sup_{x \in [0,1]} |f(x) - f_n (x)| \to 0 ,(n \to \infty)$.
I don't think it would be to difficult to show that $f \in C([0,1])$ after that and then I would have shown that $C([0,1])$ is a complete metric space. Any help is greatly appreciated.