Algebraic properties of sup and inf

Let $E,F\subseteq\mathbb{R}$ be nonempty and bounded above/below where needed, and let $c\in\mathbb{R}$.

Order properties:

• If $E\subseteq F$ and both are bounded above, then $\sup E\le \sup F$ (see supremum).
• If $E\subseteq F$ and both are bounded below, then $\inf E\ge \inf F$ (see infimum).

Translation:

• If $E+c=\{x+c:x\in E\}$, then $\sup(E+c)=\sup E + c,\qquad \inf(E+c)=\inf E + c.$

Scaling:

• If $\lambda\ge 0$ and $\lambda E=\{\lambda x:x\in E\}$, then $\sup(\lambda E)=\lambda\,\sup E,\qquad \inf(\lambda E)=\lambda\,\inf E.$
• If $\lambda<0$, then $\sup(\lambda E)=\lambda\,\inf E,\qquad \inf(\lambda E)=\lambda\,\sup E.$

In particular,

Finite unions:

• If $E$ and $F$ are bounded above, then $E\cup F$ is bounded above and $\sup(E\cup F)=\max\{\sup E,\sup F\}.$

Similarly, if bounded below then

Here max and min denote the maximum and minimum of a finite set.