Sandbox

Problem 1. We say that $m$ is the maximum of a set $A$ if $m\in A$ and $m\geq a$ for all $a\in A$.

Suppose a set $A$ of real numbers has a maximum, call it $m$. Show that $m$ is also the supremum of $A$.

Problem 2. Let $A=\{x\in\mathbb{Q}\ |\ x^2\leq 3\}$. Show that A is bounded from above, but that $A$ has no maximum.