In my homework I am asked to show that:

1) $\displaystyle \| - \|_{C^1} $ is a norm on $\displaystyle C^{1} $ [a,b] where $\displaystyle \| - \|_{C^1} $ is defined by $\displaystyle \| f \|_{C^1} $ = $\displaystyle \| f \|_{\infty} + \| f' \|_{\infty} $ (the second is meant to be 'f dash' i.e. the derivative of f )

2) $\displaystyle C^{1} $ [a,b] is complete in this norm.

My attempt so far:

1) Fairly easy, just show that the norm satisfies the three conditions in the definition of a norm.

2) This is what I'm stuck with.

$\displaystyle \left( C^1[a,b] , \| - \|_{C^1} \right) $ is complete if :

Given a Cauchy sequence $\displaystyle (f_{n}) $ in $\displaystyle \left( C^1[a,b] , \| - \|_{C^1} \right) \exists f $

such that the limit of the sequence $\displaystyle (f_{n}) $ is $\displaystyle f. $

Choose a sequence of functions $\displaystyle (f_{n}) $ in

$\displaystyle \left( C^1[a,b] , \| - \|_{C^1} \right) $ converging uniformly to $\displaystyle f:[a,b] \rightarrow \mathbb{R} $

such that $\displaystyle f \in C^1[a,b] $ . Then by definition $\displaystyle \left( C^1[a,b] , \| - \|_{C^1} \right) $ is complete.

PROBLEM: Is there any reason that there is such a sequence? Obviously I would have to show somehow why such a sequence exists ... and unfortunately I don't know how to do this because in my notes there isn't any examples to show why, for example, any finite-dimensional normed vector space is complete over $\displaystyle \mathbb{R}$.

N.B. Sorry about the format, it's the first time I've used latex!