Cauchy Sequences
A Cauchy sequence is a sequence of numbers where the terms become arbitrarily close to each other as the sequence progresses, regardless of whether the sequence converges to a limit. This concept is fundamental in real analysis and metric spaces, providing a criterion for convergence without requiring knowledge of the limit itself. It is named after the French mathematician Augustin-Louis Cauchy and is essential for defining completeness in mathematical spaces.
Developers should learn about Cauchy sequences when working in fields requiring rigorous mathematical foundations, such as numerical analysis, machine learning algorithms, or scientific computing, to understand convergence properties and error bounds. It is particularly useful in implementing iterative methods, analyzing algorithm stability, or developing proofs in theoretical computer science, ensuring that sequences behave predictably in infinite or continuous contexts.