Extended real numbers ordering of infinity

Question

Consider the extended real numbers, $\overline{\mathbb{R}}$.

Show that for any $A\subseteq B\subseteq \overline{\mathbb{R}}$, we have

$infB\leq infA$

My attempt:

The first case: Suppose $B=\varnothing$ then $infB=\infty$ and $infA=\infty$.

But, how can I then conclude that $infB\leq infA$? This doesn't make alot of sense. Is there a particular convention im not aware of?

Answer 1

$\overline{ \Bbb R}$ is a partial order, so for all its elements $x$ we have $x \le x$, in particular for $x=\infty$. No problem.