09 - Minimum and Maximum

Maximum

Let $S\subseteq \mathbb{R}$ be non-empty
Take $M\in \mathbb{R}$ then $M$ is the maximum of $S$ if

1. $M\in S$ ($M$ is an element of $S$)
2. $s\le M$ for alll $s\in S$ ($M$ is an upper bound for $S$)

TLDR maximum if $M=\sup S\in S$

Useful propery

Let $S\subseteq \mathbb{R}$ be non-empty and bounded above so by completeness axiom then $\sup S$ exists

Then $S$ has a maximum if and only if $\sup S\in S$
Also, if $S$ has a maximum then $\max S=\sup S$

Minimum

Let $S\subseteq \mathbb{R}$ be non-empty
Take $m\in \mathbb{R}$ then $m$ is the minimum of $S$ if

1. $m\in S$ ($m$ is an element of $S$)
2. $s\ge m$ for alll $s\in S$ ($m$ is an lower bound for $S$)

TLDR minimum if $m=\inf S\in S$