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.