Supremum and Infimum Algebra

Supremum/infimum algebra: Let $A,B \subseteq \mathbb{R}$ be nonempty subsets . Define $A+B=\{a+b:\ a\in A,\ b\in B\}$ and, for $c\in\mathbb{R}$, $cA=\{ca:\ a\in A\}$. Assume the displayed quantities are finite (for example, by assuming the relevant bounded above or bounded below hypotheses). Then:

• $\sup(A+B)=\sup(A)+\sup(B)$ and $\inf(A+B)=\inf(A)+\inf(B)$.
• If $c\ge 0$, then $\sup(cA)=c\,\sup(A)$ and $\inf(cA)=c\,\inf(A)$.
• If $c\le 0$, then $\sup(cA)=c\,\inf(A)$ and $\inf(cA)=c\,\sup(A)$.
• Writing $-A=\{-a:\ a\in A\}$, one has $\sup(-A)=-\inf(A)$ and $\inf(-A)=-\sup(A)$.

These rules are frequently paired with the supremum approximation lemma to turn statements about suprema and infima into $\varepsilon$-estimates.