Properties of the supremum and infimum.

Question

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 $S$?

Answer 1

• 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.

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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 $X$.

Answer 2

$\Bbb R$ 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 $M=\sup S$.