Extension: Infimum and Supremum

A supremum is a fancy word for the smallest number $x$ such that for some set $S$ with elements $a_1, a_2, ... a_n$ we have $x \ge a_i$ for all i.

In other words, the supremum is the biggest number in the set.

If there is an "Infinite" Supremum, it just means the set goes up to infinity (it has no upper bound).

Recall from the topic on infinite sets the definition of the supremum and infimum of a set $X\subseteq\mathbb{R}$.

Definition: Let $X\subset\mathbb{R}$. The supremum of $X$, denoted by $\sup X$, is the smallest value $S\in\mathbb{R}$ such that $S\ge x$ for every $x\in X$. If no such $S$ exists, then we define $\sup X=+\infty$ and say that $X$ is unbounded. If $X=\emptyset$, we set $\sup X=-\infty$.

The infimum of $X$, denoted by $\inf X$, is the largest value $s\in\mathbb{R}$ such that $s\le x$ for every $x\in X$. If no such $S$ exists, then we define $\inf X=-\infty$ and say $X$ is unbounded.

The supremum is the least upper bound and is sometimes also denoted by lub or l.u.b. Similarly, the infimum is the greatest lower bound and is sometimes denoted by glb or g.l.b. We have the relation $\sup X=-\inf(-X)$, where $-X:=\{-x:x\in X\}$.

Intuitively, the supremum and infimum are like the maximum and minimum. In fact, they are generalizations of maximum and minmum. As discussed in the topic on infinite sets, a set $X\subseteq\mathbb{R}$ may not contain a maximum or minimum. For example, the set $\{1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots\}$ does not have a minimum. It does, however, have an infimum of zero. We will talk about existence of a supremum and infimum below.

Least upper bound property: Every non-empty bounded set $X\subset\mathbb{R}$ has a least upper bound (supremum).

This gives us the existence of supremums and infimums that we want. If we have some upper bound, then we know there is a least upper bound.

Note that this does not hold for sets in $\mathbb{Q}$. Consider the bounded sequence $\{1,1.4,1.41,1.414,\ldots\}$, which writes out the decimal expansion of $\sqrt{2}$. This sequence is in $\mathbb{Q}$. But (as a set), its least upper bound is $\sqrt{2}$ which is not in $\mathbb{Q}$.

It turns out the least upper bound property is really important for sequences, which is why we work with $\mathbb{R}$ instead of $\mathbb{Q}$.

If $\sup X=+\infty$, then there is no ``maximum'' value in $X$. That means that for every $M\in\mathbb{R}$, there is an $x\in X$ with $x>M$. Otherwise, $M$ would be an upper bound for $X$.