# $(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers. [duplicate]

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $$x,y$$ and taking $$0, is it true that

$$(x+y)^r \le x^r+y^r?$$

In all the examples I considered, this turns out to be true. Moreover, to try to give a proof, does it suffices to prove that the function $$f(z)=(1+z)^r-1-z^r$$ is non-positive when $$z \in (0,1]$$ (i.e. taking $$x>y$$ and dividing by $$x$$?