Function inequality : $h(x)=2x$

Question

I have a function for which: $h(x) \leq 2x$ and $h(x+y)\leq h(x)+h(y)$. I must prove that $h(x)=2x$. I tried show that $h(x)\geq 2x$, but with no effects. Do you have any hint?

Answer 1

Let $x=0$ in the first inequality and $y=0$ in the second inequality. This gives $h(0)\le 0$ and $h(x)\le h(x) + h(0)$. Therefore, $h(0)=0$.

Now, put $y=-x$ in the second inequality and use the first inequality for $x$ and $-x$ to obtain $$0 = h(0) \le h(x)+h(-x) \le 2x -2x =0.$$

Therefore, all the used inequalities are equalities and $h(x) = 2x$, as desired.