The Correct Answer is Option A: A Cauchy sequence is convergent if it is a sequence of real numbers.

A Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any positive tolerance (epsilon), there exists a term in the sequence such that all subsequent terms are within epsilon distance of it.

To understand why a Cauchy sequence is convergent if it is a sequence of real numbers, let's break down the concepts and properties involved.

1. Definition of a Cauchy Sequence:
A sequence (x_n) is said to be a Cauchy sequence if for any positive tolerance (epsilon), there exists a positive integer N such that for all m, n ≥ N, |x_n - x_m| < />

2. Convergence of a Sequence:
A sequence (x_n) is said to be convergent if there exists a real number L such that for any positive tolerance (epsilon), there exists a positive integer N such that for all n ≥ N, |x_n - L| < />

3. Connection between Cauchy Sequences and Convergent Sequences:
In a complete metric space, every Cauchy sequence is convergent. The set of real numbers is a complete metric space, which means that every Cauchy sequence of real numbers converges to a real number.

4. The Real Numbers as a Complete Metric Space:
The real numbers have a property known as the completeness property, which states that every Cauchy sequence of real numbers converges to a real number. This property is a fundamental concept in real analysis and is used to define the real numbers as a complete metric space.

5. Examples:
a) Consider the sequence (1, 1.4, 1.41, 1.414, 1.4142, ...), which represents the decimal approximations of the square root of 2. This is a Cauchy sequence of irrational numbers that converges to the irrational number √2.

b) On the other hand, consider the sequence (1, 1.9, 2.8, 3.7, ...), which represents a sequence of rational numbers increasing by 0.9. This is a Cauchy sequence of rational numbers that does not converge to a rational number. It diverges to positive infinity.

Conclusion:
Based on the properties and definitions of Cauchy sequences and convergence, it can be concluded that a Cauchy sequence is convergent if it is a sequence of real numbers. The completeness property of the real numbers ensures that every Cauchy sequence of real numbers converges to a real number.