Properties of the supremum and infimum. Consider the set $S=\{x\in\mathbb{R}:x^{2}<2\}$. Is it true that $\sup{(S)}$ must also be a real number? If so, what would be the proof strategy?? Is this also true for the infimum of $S$?

$\Bbb R$ has the least upper bound property. Which says if $S\subset\Bbb R$ is a nonempty, set which is bounded above, then there is a real number $M$ such that $M=\sup S$.
