Show that $\lim_{x\rightarrow\infty} f(x) = \infty$ if $f$ is a convex function such that $ \lim_{x\rightarrow -\infty}f(x) = −\infty$

Question

I need to prove the theorem

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a convex function such that $ \lim_{x\rightarrow -\infty}f(x) = −\infty$. Show that $\lim_{x\rightarrow\infty} f(x) = \infty$

I can see clearly why this must hold by doing a graph where $f(x)$ converges to a constant when $x\rightarrow\infty$. But I have no idea of how to do a formal proof of this.

Answer 1

You use the fact that since $\frac{1}{2}f(x) + \frac{1}{2}f(-x) \ge f(0)$ you have that $\frac{1}{2}f(x) \ge f(0)-\frac{1}{2}f(-x)$. Then $\lim f(x) \ge 2f(0)-f(-x) = +\infty$