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 $S$?

*

*Is it true that $\sup S$ must also be a real number?


Let $S$ be a subset of $X$. By definition, the supremum is the least upper bound of the set $S$, and any upper bound lies in $X$. So $\sup S\in X$. In your case $X=\Bbb R$.


*

*If so can you refer to a proof?


The statemente follows from the definition; see above answer.


*

*Is this also true for the infimum of $S$?


The infimum is defined by $\inf S:=-\sup(-S)$, where $-S:=\{-x:x\in S\}$. So the answer is yes; see the first answer.


Definition 1 (Upper bound). Let $E$ be a subset of $X$, and 
  let $M\in X$. We say that $M$ is an upper bound for $E$, iff we have $x\le M$ for every element $x$ in $E$.

The relation $\le$ is a order relation for the set $X$.
