Twice differentiable function to infinity Let $f: \mathbb R \to \mathbb R$ be twice differentiable function, to which both $f'(x) > 0$ and $f''(x) > 0$ for all $x \in \mathbb R$. Show that $\lim_{x\to\infty}$ $f(x) = \infty$.
Tried using the definitions of differentiation but got nowhere.

Suppose that $f'(0)=a>0$. Since $f''(x)>0$, then $f'(x)>a$ for all $x>0$ (otherwise by MVT there is a point where $f''(x)\le 0$). Then $f(x)\ge ax+f(0)$ for all $x\ge 0$ - to argue this just consider $g(x)=f(x)-ax-f(0)$ and draw out a contradiction assuming $g<0$ for some $x$. Then since it's plainly true that $ax+f(0)\to \infty$ ($a>0$) you have the result
