Cauchy Sequences in Real Numbers

A Cauchy sequence is a sequence of real numbers that becomes arbitrarily close to each other as the sequence progresses. In other words, for any positive real number ε, there exists a positive integer N such that for all m, n > N, |am - an| < />

Convergence of Cauchy Sequences
A sequence is said to be convergent if it has a limit. In the case of Cauchy sequences, every Cauchy sequence of real numbers is convergent. This means that option 'A' - convergent - is the correct answer.

Proof of Convergence
To prove that every Cauchy sequence of real numbers is convergent, we can use the completeness property of the real numbers. The completeness property states that every Cauchy sequence in the real numbers converges to a real number.

Completeness Property of Real Numbers
The completeness property can be stated as follows: Every non-empty set of real numbers that is bounded above has a supremum (or least upper bound). Similarly, every non-empty set of real numbers that is bounded below has an infimum (or greatest lower bound).

Using the Completeness Property
To prove that every Cauchy sequence of real numbers is convergent, we can use the completeness property as follows:

1. Let (an) be a Cauchy sequence of real numbers.
2. Since (an) is a Cauchy sequence, it is bounded.
3. By the completeness property, the set of all terms of the sequence (an) that are less than or equal to M has a supremum, denoted by L.
4. We claim that L is the limit of the sequence (an).
5. To prove this, we need to show that for any positive real number ε, there exists a positive integer N such that for all n > N, |an - L| < />
6. Since L is the supremum of the set of terms less than or equal to M, we can choose N such that |an - L| < ε/2="" for="" all="" n="" /> N.
7. Additionally, we can choose N such that |an - am| < ε/2="" for="" all="" n,="" m="" /> N since (an) is a Cauchy sequence.
8. Combining these two inequalities, we have |an - L| = |(an - am) + (am - L)| ≤ |an - am| + |am - L| < ε/2="" +="" ε/2="" />
9. This shows that for any positive real number ε, there exists a positive integer N such that for all n > N, |an - L| < />
10. Therefore, L is the limit of the Cauchy sequence (an).

Conclusion
Every Cauchy sequence of real numbers is convergent. This is due to the completeness property of the real numbers, which guarantees that every bounded Cauchy sequence has a limit. Therefore, option 'A' - convergent - is the correct answer to the given question.