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. However, it differs in that a Cauchy sequence only refers to the tail of the sequence and not to some (usually unknown) limit [1].
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 [2]: A sequence of real numbers is convergent if and only if it is Cauchy.
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].
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).