The First Lesson in Analysis
Definition of a Real Number
updated 1jul13 and 16mar16

\begin{document}
\maketitle

\section{Introduction}
In order to practice the \textit{analysis of real numbers},
which is what the theory behind the differential and integral calculus is
properly  called, we obviously need to know just what a real number is. This
turns out to be more difficult than might be thought. The high school answer
that a real number is a finite or infinite decimal is fine for most practical
purposes, but won't do for mathematics.

Our textbook (p. 256) uses an unintuitive but easily stated definition. But it
is quite difficult to work with in practice because it is so far removed
from the concept of a decimal expansion. So, after giving the
"book definition" we shall give two more: the concept of a
\textit{Toeplitz Sequence} generalizes decimals, and the concept of
a \textit{Cauchy Sequence} generalizes this some more.
In return for the extra effort of learning what they all say,
you may use any one of them to solve homework
problem. This simplifies many exercises in the book. For, in a particular
case you may wish to use one of the three definition in the hypothesis of
a theorem you're proving, and different one for the conclusion.
By studying all of these definitions you'll also get a good idea of what a
real number really is, but you will have to wait for a proper analysis course
to see why all these definitions are equivalent.

Note that, having invested considerable effort to master logical notation
in this course, we shall now use it to give precise definitions. The textbook
translates these into more or less common English. You should both into your
Journal, with the paraphrase in your own words, if possible.

\section{The Least Upper Bound form of the Completeness Axiom}
This section of the text begs the question of the existence of real numbers,
indeed, I could not find a clear definition of the reals in the index of the
book. But it doesn't matter, because we can start by saying that all rational
numbers are also real numbers. We then define some properties of real numbers,
whatever they are. Finally, when we know what we want real numbers to be, we
use these properties to define them. No fear, the logicians know how to
do this without seeming circularity. But their exposition would not go over
well in this course.

\subsection{Definition of an Upper Bound }
For $S \subset \mathbb{R}$ and number $u \in \mathbb{R}$,  we say that
$u$ is an \textit{ upper bound} for $S$, written $u \in ub(S)$, if

\begin{eqnarray*}
(\forall x) (x \in S &\Rightarrow& x \le u ) \\
\mbox{Which shortens to } (\forall x \in S)&&(x \le u)\\
\mbox{and whose negation is}\\
(\exists x) (x \in S &\and &x > u) \\
\mbox{Or, simply } &&\\
(\exists x\in S) &  & (x > u) \\
\end{eqnarray*}

The \textit{ least upper bound } of a set $S$, written $\ell = lub(S)$,
is an upper bound, $\ell \in ub(S)$, which is less than every other
upper bound:

\begin{eqnarray*}
(\forall u \in ub(S))  & & (\ell \le u) \\
\mbox{And, more precisely} &&\\
(\forall u) (u \in ub(S)  & \Rightarrow& \ell \le u) \\
\mbox{Whose negation becomes}&&\\
\neg (\forall u) (u \in ub(S)  & \Rightarrow& \ell \le u) \\
(\exists u)(u \in ub(S) & \and & u < \ell) \\
\mbox{Or, simply } &&\\
(\exists u \in ub(S)) & & (u < \ell) \\
\end{eqnarray*}

Question 1.

Rewrite the definition of least upper bound as a single logical
statement without reference to the intermediate concept  $ub(S)$.

\textbf{Least Upper Bound Axiom}\\
Every non-empty subset of reals that has an upper bound
also  has a least upper bound.

A similar definition applies to the set $lb(S)$ of \textit{ lower bounds}
a set $S \subset \mathbb{R}$, and its \textit{ greatest lower bound },
$glb(S)$.

Question 2.

State the completeness axiom using lower bounds.

Next, show that these two concepts are equivalent, using the \textit{ negative
of a set of numbers}
$\bar{S} = \{ -s | s \in S \}$.

Question 3.

Show that $\overline{ub(S)} = lb(\bar{S})$.

Then complete the argument thus.

Question 4.

Show that $glb(S) = -lub(\bar{S})$.

Note that subset might have a \textit{ maximum },
written $m=max(S)$, and defined by
$m \in S \ \wedge \ (\forall s \in S)(s \le m)$.

Question 5.

Show that if $m = max(S)$ then $m = lub(S)$.

Question 6.

Show by a counterexample that the converse is not always true.

As similarly defined concept of a \textit{ minimum} of $S$ is similarly defined,
and written $min(S)$.

\subsection{Examples in the Text}
[See the textbook p 257 for now.] The authors use the modern terms
\textit{supremum, infimum} for \textedit{lub, glb}. The examples are
more sophisticated and harder to understand than the ones we give here.

\subsection{Definition of the Reals using the LUB Axiom}
Since, as the text points out, the rational numbers are \textit{ incomplete},
in the sense that $S = \{ x \in \mathbb{Q} | x^2 \le 2 \}$ is bounded
above, but there is no rational $lub(S)$, we supply one using the concept
of a \textit{ Dedekind Cut} (google it).

\textbf{ Definition}
A real number is an ordered pair of
\textbf{ nonempty subsets} $A,B \subset \mathbb{Q}$
such that
\begin{eqnarray*}
\mbox{1. } (\forall a, b \in \mathbb{Q})( a\in A \wedge b \in B &\Rightarrow& a < b) \\
\mbox{2. } \neg(\exists max(A)) &\wedge & \neg(\exists min(B) ) \\
\mbox{3. }A\cup B & = & \mathbb{Q} \mbox{    minus at most one rational number} \\
\end{eqnarray*}

The real number so defined is called the \textit{Dedekind Cut} for $A,B$. When there really
is a rational number $r$ between $A$ and $B$, then is simultaneously the
$lub(A)$ and the $glb(B)$. This takes a little effort to prove, and you should put
its proof into your journal. But when $A \cup B = \mathbb{Q}$, then the cut defines
an irrational real number, which is also the $lub(A)$ and $glb(B)$.

\subsection{Examples}

A rational number, say $r=42$, may now be redefined by the Dedekind cut $(A,B)$ where
$A = \{a \in \mathbb{Q} | a < 42 \}$ and $B = \{b \in \mathbb{Q} | b > 42 \}$.
This shows that the rationals have an injection into the reals (the set of
Dedekind cuts of the rations). So we consider the rationals,
$\mathbb{Q} \subset \mathbb{R}$, a subset of the reals. Recall (where) we
made a similar definition that includes the natural numbers among the rationals.

For the irrational number $x=\sqrt{2}$, the defining Dedekind cut would consist
of the negative rationals, zero, and positive rationals with $a^2 < 2$ for $A$,
and its complement $B = \mathbb{Q}- A$. You should rewrite the exposition in
the textbook concerning the $\sqrt{2}$ so as to verify that all four conditions
of a Dedekind cut are satisfied.

\subsection{Historical Note}
Although Dedekind lived in the 19th century, the concept the Greeks used instead
of real numbers is essentially equivalent to Dedekind's concept. Greeks came nowhere
near the Least Upper Bound principle. This is a topic in our advanced course
on the history of the Calculus, MA406.

Having defined reals, we should have the courtesy of proving at least some of
the properties we expect the reals to have. In other words, that this is a good
definition. Unfortutanely, this is very difficult when the Completeness of the
reals is defined in terms of the Least Upper Bound principle. One exception might
be to show from this definition that the Least Upper Bound Axiom actually holds
for this definition. Consider this a challenge problem for extra credit.

\subsection{Seminar of the Least Upper Bound Axiom}
In a separate document we give a representative collection of problems to
solve for you to really understand the above. The text is inaquedate for this
purposes. We shall treat the solutions to these problems in a Seminar.

\section{Definition of the Reals by Toeplitz Sequences }
The students of my 2006 class wrote up a good set of class notes.
See  class notes  together with
a detailed exposition of the example, based on compound interest, of
$lim_{n\rightarrow \infty} ( 1 + \frac{1}{n})^n = \e$. This material
in not in the textbook but constitutes an integral part of this course.

\section{Cauchy Sequences}
[To be completed]
\subsection{Motivation from Convergence of Sequences}
\subsection{Definition of the Reals by Cauchy Sequences}

\section{Exampls of how to use these three definitions}
[To be completed]

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