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The Axiom of Completeness (Part 2)

MATH 4630 · Chapter 1: The Real Numbers

In the previous handout, we stated the following axiom:

Axiom 3: Every nonempty subset of $\mathbb{R}$ that is bounded above has a least upper bound in $\mathbb{R}$.

In this handout, we will explore Axiom 3 in detail. Let us begin with some important definitions.

Bounded Sets

A set $A\subseteq\mathbb{R}$ is bounded above if there exists $b\in\mathbb{R}$ such that

$$a \le b \text{ for all } a \in A.$$

Such a number $b$ is called an for $A$.

A set $A\subseteq\mathbb{R}$ is bounded below if there exists $\ell\in\mathbb{R}$ such that

$$\ell \le a \text{ for all } a \in A.$$

Such a number $\ell$ is called a for $A$.

For each set below, complete the table.

Set	Bounded Above?	Bounded Below?	One Upper Bound	One Lower Bound
(i) $A=(0,1)$
(ii) $B=\{x\in\mathbb{R}:x^2<4\}$
(iii) $C=\{x\in\mathbb{R}:x\ge 3\}$
(iv) $D=\left\{\frac{1}{n}:n\in\mathbb{N}\right\}$
(v) $E=\{x\in\mathbb{R}:x<5\}$

Supremum and Infimum

A real number $s$ is the least upper bound or supremum of $A$ (written $s=\sup A$) if:

$s$ is an upper bound for $A$: for all $a \in A$, $a \le s$.
$s$ is the least such bound: if $b$ is any upper bound for $A$, then $s \le b$.

A real number $\ell$ is the greatest lower bound or infimum of $A$ (written $\ell=\inf A$) if:

$\ell$ is a lower bound for $A$: for all $a \in A$, $\ell \le a$.
$\ell$ is the greatest such bound: if $m$ is any lower bound for $A$, then $m \le \ell$.

Let $A\subseteq\mathbb{R}$ be a nonempty set that is bounded above. Suppose $s_1$ and $s_2$ are both supremums of $A$. Prove that $s_1 = s_2$.

The supremum of a set .

If a set is , then it has no supremum (e.g., $\mathbb{N}$).
The empty set has supremum. By convention, one sometimes writes $\sup\varnothing=-\infty$, but this is not a real number and is outside our definition.

For each set below, determine $\sup A$ and $\inf A$ (if they exist) and whether they belong to the set.

Give an example from the list above where:

Maximum and Minimum

In general, $\sup A$ and $\inf A$ need not belong to the set $A$.

If $\sup A\in A$, then $\sup A$ is called the maximum of $A$, and we write $\max A=\sup A$.
If $\inf A\in A$, then $\inf A$ is called the minimum of $A$, and we write $\min A=\inf A$.

An Equivalent Characterization of Supremum

Let $A\subseteq\mathbb{R}$ be nonempty and bounded above, and assume $s$ is an upper bound for $A$. Then $s=\sup A$ if and only if:

For every $\varepsilon>0$, there exists $a\in A$ such that $s - \varepsilon < a \le s$.

For any $\varepsilon > 0$, the interval $(s-\varepsilon, s]$ must contain at least one point from $A$.

Takeaway: No matter how small $\varepsilon$ is, $A$ must come within $\varepsilon$ of $s$ from below.