In my homework I am asked to show that:

1) is a norm on [a,b] where is defined by = (the second is meant to be 'f dash' i.e. the derivative of f )

2) [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.

is complete if :

Given a Cauchy sequence in

such that the limit of the sequence is

Choose a sequence of functions in

converging uniformly to

such that . Then by definition 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 .

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