User:Mathemajor/Contractive sequences

In mathematics and, in particular, analysis a sequence $$\lbrace x_n \rbrace$$ in a metric space $$(M, d)$$ is said to be contractive if the distance between consecutive terms in the sequence shrinks in a predictable manner as one observes terms further into the sequence. The word "contractive" stems comes from the fact that, formally, a contractive sequence $$\lbrace x_n \rbrace$$ is a contractive function $$f: \mathbb{N} \to M$$ such that $$f(n) = x_n$$ for all $$n \in \mathbb{N}$$. Due to satisfying several desirable properties (including being Cauchy), observing that a sequence is contractive immediately provides insights into its behavior. Beyond analysis, contractive sequences are useful and arise naturally in the general theory that allows computer scientists to generate a large class of functions efficiently via iterated function sequences.