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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeb81122-6fbf-4c63-aaf5-71689ff4fea7%2F4748a0d7-8006-4859-8b1e-c37a6ee25e3e%2Fxp5mast_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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