Let $\displaystyle f:\mathbb{R}\to \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies

$\displaystyle f (x + y) \leq yf (x) + f (f (x))$

for all real numbers $\displaystyle x$ and $\displaystyle y$. Prove that $\displaystyle f(x)=0$ for all $\displaystyle x\leq 0$.