Proving the Lp norm is a norm.

I want to prove the $L_p$ norm on continuous functions is in fact a norm. I have proven definitiveness and homogeneity but am struggling with the triangle inequality. I am using the fact that if the closed unit ball is convex then the triangle inequality holds.

Attempt so far:

Let $f$,$g$ be two functions in the closed unit ball, let $h=af+(1-a)g$, for some $a \in [0,1]$.

Then the $L_p$ norm of $h$ is

$$\left(\int|af+(1-a)g|^p\right)^{1/p}$$

I think i want to try and decompose this into 2 components, one containing the integral of $f$, one containing the integral of $g$ but am unsure how to do this given the power of $p$.Thanks in advanceDan