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.

Note that $f''$ is positive and hence $f'$ is strictly increasing and hence $f'(x) $ tends to a limit $L$ or to infinity. Moreover if it tends to $L$ then $L\geq f'(0)>0$. Again using the same logic we can see that either $f(x) $ tends to $M$ or to infinity. If $f(x) \to M$ then $f(x+1)-f(x)=f'(\xi)$ tends to $0$ which contradicts that $f'$ tends to $L>0$ or to infinity. It follows that $f(x) \to\infty$ as $x\to\infty$. 
