## Real analysis – Basics III

— 3. Completeness Axiom —

As was said in the previous post the description of the set of real numbers made so far allows one to do almost everything but to formally differentiate the set ${ \mathbb{Q}}$ and the set ${ \mathbb{R}}$. To do that we need one more axiom; but first let us introduce some auxiliary notions.

For ${ a,b \in \mathbb{R}}$ one usually writes ${ \lbrack a,b\rbrack}$, ${ \lbrack a,b \lbrack}$, ${ \rbrack a,b\rbrack}$ and ${ \rbrack a,b\lbrack}$ for the real numbers ${ x}$ for which the following conditions are verified:

• ${ a \leq x \leq b}$
• ${ a \leq x < b}$
• ${ a < x \leq b}$
• ${ a < x < b}$

${ \lbrack , \rbrack}$ is called a closed interval and ${ \rbrack , \lbrack}$ is called an open interval.

We can also have another kind of intervals. These intervals are said to be infinite and have the form ${ \lbrack a, \infty \lbrack}$, ${ \rbrack a, \infty \lbrack}$, ${ \rbrack -\infty,a\rbrack}$, and ${ \rbrack -\infty, a\lbrack}$. They represent respectively:

• ${ x \geq a}$
• ${ x > a}$
• ${ x \leq a }$
• ${ x < a }$
 Definition 5 Lets ${ \mathrm{K}\subset \mathbb{R}}$ and ${ a,b \in \mathbb{R}}$. We’ll say that the real ${ b}$ is an upper bound of ${ \mathrm{K}}$ if and only if (for now on written as iff) ${ \forall x \in \mathrm{K} \,x \leq b}$. We’ll also say that ${ a}$ is a lower bound of ${ \mathrm{K}}$ iff ${ \forall x \in \mathrm{K} \,a\leq x}$. We’ll say that ${ \mathrm{K}\subset \mathbb{R}}$ is bounded from above if it has at least one upper bound, bounded from below if it has at least one lower bound, and ${ \mathrm{K} \subset \mathbb{R}}$ is said to be bounded if it has an upper and a lower bound. Finally a set is said to be unbounded if it isn’t bounded.
 Definition 6 Let ${ \mathrm{K} \subset \mathbb{R}}$. If ${ \exists x \in \mathrm{K} \land \forall y \in \mathrm{K} x\geq y}$, then ${x}$ is the maximal element of ${ \mathrm{K}}$ and we denote it by ${\max \mathrm{K}}$ If ${ {\mathrm K} \subset \mathbb{R}: \exists u \in {\mathrm K} \land \forall v \in {\mathrm K} \,u \leq v}$, ${ u}$ is said to be the the minimal element of ${ {\mathrm K}}$ and is denoted by ${ \min {\mathrm K}}$.
 Definition 7 Let ${ {\mathrm K} \subset \mathbb{R}}$ and ${ {\mathrm V}}$ the set of all of its upper bounds (${ {\mathrm V}=\emptyset}$ iff ${ {\mathrm K}}$ isn’t bounded from above). The minimal element of ${ {\mathrm V}}$, ${ s}$, is said to be the supremum of ${ {\mathrm K}}$. ${\sup {\mathrm K}}$ Formally: $\displaystyle \forall x \in {\mathrm K}; x \leq s \ \ \ \ \ (6)$ $\displaystyle \forall v \in {\mathrm V}; s\leq v \Leftrightarrow \forall \epsilon > 0 \exists x \in {\mathrm K}: x > s-\epsilon \ \ \ \ \ (7)$

The notion of the infimum of a set can be introduced in an analogous way.

 Definition 8 Let ${ {\mathrm K} \subset \mathbb{R}}$ and ${ {\mathrm U}}$ the set of lower bounds of ${ {\mathrm K}}$. The infimum of ${ {\mathrm K}}$ is the maximal element of ${ {\mathrm U}}$. If such a a maximal element exists the infimum of ${ {\mathrm K}}$ is denoted by ${ \inf {\mathrm K}}$ and is the real number ${ r}$ such as: $\displaystyle \forall x \in {\mathrm K}; x\geq r \ \ \ \ \ (8)$ $\displaystyle \forall u \in {\mathrm U}; u\leq r \Leftrightarrow \forall \epsilon >0 \exists x \in {\mathrm K}: \quad x < r+\epsilon \ \ \ \ \ (9)$

Let us see some examples of all of these previous notions so that we can have a feel of what’s going on.

• ${\sup \lbrack a,b\rbrack = \max \lbrack a,b\rbrack = b}$
• ${\inf \lbrack a,b\rbrack = \min \lbrack a,b\rbrack = a}$
• ${\sup \rbrack a,b\lbrack = b}$
• ${\inf \rbrack a,b\lbrack = a}$

Notice that the two last sets have no maximal nor minimal element.

As a work out the reader can try to find ${\min}$, ${\max}$, ${\sup}$, and ${\inf}$ of the empty set.

And now we’ll state the Completeness Axiom and with it our basic study of the real numbers will be complete.

 Axiom 8 (Completeness Axiom) Any non-empty subset of ${ \mathbb{R}}$ with an upper bound has a real supremum.
 Theorem 9 Any non-empty subset of ${ \mathbb{R}}$ which is bounded from below has an infimum. Proof: Omitted. $\Box$
 Theorem 10 ${ \mathbb{N}}$ isn’t bounded above. Proof: Omitted. $\Box$
 Theorem 11 (Archimedean property) ${ a,b \in \mathbb{R} \land a > 0 \Rightarrow \exists n \in \mathbb{N}: na > b}$ Proof: Omitted. $\Box$

Theorem 11 tell us that in ${ \mathbb{R}}$ infinitesimals don’t exist: if we add any given quantity ( it doesn’t matter how small it is ) a sufficient number of times the value of the sum will always be greater than any other given quantity.