**Definition and Proof of Contractive Sequences**A sequence \((s_n)\) is said to be contractive if there exists a constant \(k\) with \(0 < k < 1\) such that:\[|s_{n+2} - s_{n+1}| \leq k |s_{n+1} - s_n|\]for all \(n \in \mathbb{N}\).**Problem:** Prove that every contractive sequence is a Cauchy sequence and hence is convergent.

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Answered: 1 A sequence (m) is said to be…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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1 A sequence (m) is said to be contracture if there exists a constant k with ockal such that 11m+2-Anti |sk| Anfi-^m/ for all mex. Prove that contractive sequence is a and hence is convergent. every Cauchy sequence.