Math - Supremum and Infimum

Most content of this post is from the Lecture 2 for Prof. John K. Hunter’s Math 125b and Wikimedia.

Supremum & Infimum of Sets

Definitions

Let $A \subseteq \mathbb{R}$, if $M \in \mathbb{R}$ is the smallest upper bound of $A$, i.e., for any upper bound $M'$ of $A$, we have $M \leq M'$. We call $M$ the supremum of $A$, denoting as $M =\sup A$. If $m \in \mathbb{R}$ is the largest lower bound of $A$, then we call $m$ the infimum of $A$, denoting as $m = \inf A$.

Properties

The supremum or infimum of a set $A$ is unique if exists. Moreover, if both exist, then $\inf A \leq \sup A$.

For $c \geq 0$,
$\sup (cA) = c\sup A,\qquad \inf (cA) = c \inf A$
For $c < 0$,
$\sup (cA) = c \inf A,\qquad \inf (cA) = c\sup A$

Let $A \subseteq B \subseteq \mathbb{R}$, if $\sup A$, $\sup B$ exist, then $\sup A \leq \sup B$. If $\inf A$, $\inf B$ exist, then $\inf A \geq \inf B$.

Let $A,B \subseteq \mathbb{R}$ be non-empty sets, then
$\begin{aligned} \sup (A + B) = \sup A + \sup B,\qquad \inf (A +B)= \inf A + \inf B,\\ \sup (A - B) = \sup A = \inf B,\qquad \inf (A - B) = \inf A - \sup B. \end{aligned}$

Supremum \& Infimum of Functions

Definition

Let $E \subseteq \mathbb{R}$, $f: E \mapsto \mathbb{R}$ is a function, then,
$\sup_{E} f = \sup\{f(x):x \in E\},\qquad \inf_{E} = \inf\{f(x):x \in E\}$

Properties

Let $E \subseteq \mathbb{R}$, $f,g: E \mapsto \mathbb{R}$, if $g$ is bounded from above, then
$\sup_{E} f \leq \sup_{E} g,$
if $f$ is bounded from below, then
$\inf_{E} f \leq \inf_{E} g.$

Let $E \subseteq \mathbb{R}$, $f: E \mapsto \mathbb{R}$ is bounded, $c \in \mathbb{R}$, if $c \geq 0$, then
$\sup_{E} (cf) = c \sup_{E} f,\qquad \inf_E (cf) = c \inf_E f.$
if $c < 0$, then
$\sup_E (cf) = c \inf_E f,\qquad \inf_E (cf) = c \sup_E f.$

Let $E \subseteq \mathbb{R}$, $f,g: E \mapsto \mathbb{R}$ are bounded, then
$\sup_E (f + g) \leq \sup_E f \sup_E g,\,\inf_E (f + g) \geq \inf_E f + \inf_E g.$

Let $E \subseteq \mathbb{R}$, $f,g: E \mapsto \mathbb{R}$ are bounded, then
$\mid \sup_E f - \sup_E g\mid \leq \sup_E \mid f - g \mid,\qquad \mid \inf_E f - \inf_E g\mid \leq \sup_E \mid f - g\mid.$