The Cauchy Convergence Theorem states that a real-numbered sequence converges if and only if it is a Cauchy sequence. A Cauchy sequence is a series of real numbers (sn), if for any ε (a small positive distance) > 0, there exists N, such that m, n > N implies |sn – sm| < ε. In other words, the elements of the sequence become arbitrarily close to each other as the series progresses.
The definition of a Cauchy sequence is very similar to the definition of a convergent sequence. A convergent sequence is a sequence of numbers that has a limit. which means that the terms of the sequence get closer and closer to a specific number as the terms get farther apart. However, a Cauchy sequence only refers to the tail of the sequence and not to some (usually unknown) limit [1]. This means that a Cauchy sequence can be convergent, but it does not have to be.
For example, the sequence xn = 1/n is a Cauchy sequence, but it is not convergent. The terms of the sequence get closer and closer to 0, but they never actually reach 0.
On the other hand, the sequence xn = 1/n2 is both Cauchy and convergent. The terms of the sequence get closer and closer to 0 as the terms get farther apart, and they eventually reach 0.
Two useful lemmas are associated with Cauchy convergence [2]:
- Every convergent sequence is Cauchy.
- Every Cauchy sequence is bounded.
Cauchy Convergence Criterion
The Cauchy convergence criterion is useful for proving that a given sequence is convergent without having to find a limit. The criterion states that a deterministic sequence (xn) is Cauchy if and only if it is convergent [3]. This can also be stated in reverse: a sequence of real numbers is convergent if and only if it is Cauchy [2]. This means that if we can show that a sequence is Cauchy, then we can be sure that it is convergent, even if we do not know the limit of the sequence.
The Cauchy convergence criterion does not apply to non-deterministic sequences. A non-deterministic sequence is a sequence whose terms are not determined by a specific formula or rule. For example, a sequence of coin flips is a non-deterministic sequence.
The Cauchy convergence criterion is a necessary and sufficient condition for sequence convergence. In general, “necessary” means that it must be present in order for convergence to happen and “sufficient” means that it produces the condition (i.e. that it produces convergence)[3]. The Cauchy convergence criterion is proved with the compactness theorem and the interval proof theorem [5].
The Cauchy convergence criterion is also useful for proving that a sequence is divergent. A sequence is divergent if it is not convergent. If we can show that a sequence is not Cauchy, then we can be sure that it is divergent.
References
[1] Completeness of R. Retrieved June 18, 2021 from: https://www.iiserkol.ac.in/~adg/courses/ma311/ch1.pdf
[2] Grigoryan, V. (2011). Convergence criteria for sequences. Retrieved June 18, 2021 from: http://web.math.ucsb.edu/~grigoryan/117/lecs/lec13.pdf
[3] De Cataldo, M. (2021). Cauchy Convergence Criterion. Retrieved June 18, 2021 from: http://www.math.stonybrook.edu/~mde/319S_08/testsolII.pdf
[4] Texas State. Confusion of Necessary with a Sufficient Condition. Retrieved June 18, 2021 from: https://www.txstate.edu/philosophy/resources/fallacy-definitions/Confusion-of-Necessary.html
[5] Wan, M. Study on the Sufficient Certification of Cauchy Convergence Criterion. Advances in Engineering, volume 126. 5th International Conference on Machinery, Materials and Computing Technology (ICMMCT 2017).