Cauchy sequence of rational numbers


	Cauchy sequence of rational numbers

Natural numbers

${\rm N}=\{1,\;2,\;\dots \}$

serve well for counting and ordering objects. However, it is impossible to find a solution for the equation

$x+n=0,\;\;\mbox{ where } n \in {\rm N}$

in the set of natural numbers. That leads us to finding extensions for N. The natural extension is provided by integers,

${\rm Z} = \{\dots \; -3,\; -2 ,\; -1,\; 0,\; 1,\; 2,\; 3,\; \dots \}$

The solution for the equation

$x+n=0,\;\;\mbox{ with } n \in {\rm N}$

is given by

x = -n

and it is an integer, $x \in {\rm Z}$ . The set of integers Z itself has some limitations. In particular, the equation

$n\cdot x = m ,\;\;\mbox{ where } m,\;n \in {\rm Z} \;\; \mbox{ and } n\not= 0$

does not have solutions in Z. In order to resolve this problem we also need to extend the set of integers and add to it all possible ratios,

$\frac{m}{n},\;\;\mbox{ with } n \not= 0 \mbox{ and } m,\;n \in {\rm Z}$

The resulted set is called rational numbers and denoted by Q. In other words,

${\rm Q } = \{ \frac{m}{n};\;\;m,\;n \in {\rm Z} \mbox{ and } n\not= 0 \}.$

Rational numbers Q suit wide range of various practical applications. Nevertheless, it is impossible to find in Q the solutions for

x² = p,

where p > 1 is a prime natural number; p can be divided without remainder only by 1 and itself. One can prove this statement by reductio ad absurdum. Suppose there exists a rational number $\frac{m}{n}$ such that

$(\frac{m}{n})^2 = p.$

Without loss of generality, one can assume that m and n are coprime; the largest common divisor for m and n is 1. Hence,

$m^2 = p \cdot n^2$

Since m² is divided by p without remainder then so is m. Therefore, $m=p\cdot k,$ where $k\in {\rm Z}.$ Thus, we come to conclusion that

$m^2 = p^2 \cdot k^2$

and $m^2= p \cdot n^2$ implies that

$n^2=p \cdot k^2$

It contradicts to the assumption that m and n are coprime. The obtained contradiction proves that the equation

$x^2 = p, \;\;\mbox{ where } p>1 \;\;\mbox{ is prime }$

does not have any solutions in the set of rational numbers, Q.

Now we need to extend rational numbers Q in order to be able to solve the equation x² = p with prime p > 1. The corresponding extension is called the set of real numbers and is denoted by R. To illustrate the construction of real numbers we use the following geometrical interpretation. All numbers can be associated with points on a straight line.

The distance between two numbers x and y is defined as distance between the two corresponding points and denoted as

$\vert x - y \vert .$

One can see that rational numbers are dense everywhere on the straight line. In order to justify this statement consider a point A on the interval $[0,\;1)$ that contains all points between 0 and 1 including 0 and excluding 1. Given A one can construct the sequence of the rational numbers as follows.

A₀ = 0

$A_1 = \frac{a_1}{2}$

$A_2 = \frac{a_1}{2} + \frac{a_2}{2^2}$

$A_3 = \frac{a_1}{2} + \frac{a_2}{2^2} + \frac{a_3}{2^3}$

$\;\;\;\;\;\; \vdots$

$A_n = \frac{a_1}{2} + \frac{a_2}{2^2} + \frac{a_3}{2^3}+\dots + \frac{a_n}{2^n}$

where each of the coefficients {a_i} takes only values 0 and 1. The values assigned according to the following rule

$a_n=1 \;\;\mbox{ if } \;\; A \ge A_{n-1}+\frac{1}{2^n}$

and

$a_n=0\;\;\mbox{ if } \;\; A < A_{n-1}+\frac{1}{2^n}$

where $n=1,\;2,\;\dots$

The sequence {A_n} is a Cauchy sequence in the sense of the following definition.

Definition A sequence {x_n} is a Cauchy sequence if, and only if,

$\forall \;\;\varepsilon >0 \;\;\;\exists \;\;M\in{\rm N}\;\;\mbox{ such that }\;\;\forall\;\;m>M \;\;\mbox{ and }\;\;\forall\;\;n>M\;\;$

$\mbox{ we have } \vert x_n - x_m \vert <\varepsilon$

For constructed above sequence {A_n} we have

$\vert A_m - A_n \vert =\frac{a_{n+1}}{2^{n+1}}+ \dots +\frac{a_{m}}{2^{m}}$

if we assume that m > n. Since $\forall \;\;i\in{\rm N}\;\;a_i\le 1$ we conclude that

$\frac{a_{n+1}}{2^{n+1}}+\dots + \frac{a_{m}}{2^{m}}\le \frac{1}{2^n}$

Hence, {A_n} is a Cauchy sequence. Moreover, if you draw the points corresponding the the members of this sequence {A_n} then you can see that they are getting closer and closer to the point A as n increases. We will use this property to define real numbers in the following lecture. Here, we only mention that

$0.a_1 a_2 a_3 \dots a_n \dots$

is called a binary representation for A.

Cauchy sequence of rational numbers