Prove $(x+y)^a\leq x^a+y^a$ if $0Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$
I need to prove this step for a bigger question. It should be quite basic but I just have no idea...

Let $f(x)=x^{a}$. We may assume $y\ge x$. The line $y=L(x)$ joining $(0, 0)$ and $(y, f(y))$ satisfies the following
$$
f(x)+f(y)\ge L(x)+L(y)=L(x+y)\ge f(x+y),
$$
since $f$ is concave.
