—  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.