Pre Clarkson Inequality

I am actually struggling with a seemingly totally trivial inequality.

For any non negative $a,b$ and $p > 2$:

$a^p + b^p \leq (a+b)^p$.

For natural $p$ this is obvious, but what is with the rationals or even irrationals between two integers?

The plan was to show the inequality holds for any rational $p>2$, and via continuity, it has for any real $p>2$ then.But I have no idea how to show this.

I tried it with several functions an convexity but i failed.

I hope someone has a smooth solution.