Examples of Cauchy Sequences That Do Not Converge to a Number

In the realm of mathematical analysis, a sequence is considered a Cauchy sequence if the terms of the sequence become arbitrarily close to each other as the sequence progresses. This concept is pivotal in understanding the properties of various metric spaces. While in complete metric spaces, every Cauchy sequence converges to a limit within that space, in incomplete metric spaces, it is possible for Cauchy sequences to not converge to a point within the space.

Cauchy Sequence in the Rational Numbers

Let us consider the sequence defined by the decimal expansions of the square roots of natural numbers that are not perfect squares. The sequence is given by:

a_n sqrt{n} text{ for } n 2, 3, 5, 6, 7, 8, 10, ldots

This sequence is Cauchy in the rational numbers because the terms get arbitrarily close to each other as ( n ) increases. However, the square roots of natural numbers that are not perfect squares (e.g., ( sqrt{2} ), ( sqrt{3} ), etc.) are not rational numbers, and therefore, the sequence does not converge to a number within the set of rational numbers. This exemplifies the limitations of the rational numbers as a complete space.

Cauchy Sequence in the Space of Continuous Functions

Another fascinating example is a sequence of functions defined on the interval ([0, 1]). The sequence of functions is given by:

f_n(x) x^n text{ for } x in [0, 1]

In the space of continuous functions with the supremum norm, this sequence is Cauchy. As ( n ) increases, ( f_n(x) ) approaches ( 0 ) for any ( x in [0, 1) ). Yet, at ( x 1 ), ( f_n(1) 1 ). This indicates that the limit function is not continuous on ([0, 1]). Therefore, the sequence does not converge to a continuous function within the space of continuous functions on ([0, 1]).

Cauchy Sequence in the Space of Rational Numbers Approximating Irrational Numbers

A third example involves a sequence of rational numbers approximating an irrational number. The sequence is defined by:

a_n frac{1}{n} text{ for } n 1, 2, 3, ldots

This sequence is Cauchy since the terms get arbitrarily close as ( n ) increases. However, if we consider the sequence of rational approximations to ( pi ), such as ( 3.14, 3.141, 3.1415, ldots ), the sequence is Cauchy in the rational numbers but converges to ( pi ), which is not a rational number. This situation illustrates the inherent incompleteness of the rational numbers and the need for a more comprehensive space.

Summary

In conclusion, Cauchy sequences can exist in incomplete metric spaces, such as the rational numbers, or in the space of continuous functions on a closed interval, where they do not converge to a limit within the space. Understanding these concepts is crucial for delving into the intricacies of mathematical analysis and topology.