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

Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.

So let $\displaystyle f_{n}$ in $\displaystyle C^{1} $ be a sequence of functions and suppose that $\displaystyle f_{n} $ is Cauchy.

So for every $\displaystyle \varepsilon > 0$ there is an N such that $\displaystyle n, l \geq N \implies ||f_{n} - f_{l}|| < \varepsilon$.

Need to show that $\displaystyle f_{n}$ is convergent.

Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?