Let $A\subseteq\mathbb R$.
Is there a nice proof of the inequality $\displaystyle\inf_{a\in A} |a|\le|\inf_{a\in A} a|$?
The only proof I know is, though not very difficult, annoying because it requires case analysis (three cases, depending on where $\inf a$ is in comparison to $-\inf_{a\ge 0} a$ and $0$). I wonder if I'm overlooking a nice calculational proof.
Although we still have to separate cases, maybe this is simple enough. We have two cases:
1) If $A$ has some element $a_0\leq 0$, then $\inf_{a\in A}a\leq a_0\leq0$, thus $$\inf_{a\in A}|a|\leq |a_0|=-a_0\leq -\inf_{a\in A}a\leq |\inf_{a\in A}a|.$$
2) If all elements of $A$ are positive, then $$\inf_{a\in A}|a|=\inf_{a\in A}a=|\inf_{a\in A}a|.$$