I do understand what a convex function is and I can see geometrically that following statement is true:
$(1)$ If for a given convex function $f$ which is differentiable over $\mathbb{R}$, if $f'(x_0)>0$ for some $x_0$ then $\lim_{x \to \infty}{f(x)} = \infty$.
I see it like this: if the derivative $f'(x_0)>0$ then the slope is increasing all the time as long as you approach infinity from $x_0$ due to the definition of convex function:
$$f(tx_0 + (1-t)x_1 \leq tf(x_0) + (1-t)f(x_1)$$
In other words, the secant is above the curve all the time. But how do I prove the statement $(1)$?