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Let (n) and (yn) be Cauchy sequences. Using only the definition of Cauchy (and not using Theorem 2.6.4, or Algebraic Limit Theorems) prove that (xn+yn) is Cauchy.

Let (n) and (yn) be Cauchy sequences. Using only the definition of Cauchy (and not using Theorem 2.6.4, or Algebraic Limit Theorems) prove that (xn+yn) is Cauchy.

Advanced Engineering Mathematics
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:** A sequence \((a_n)\) is called a *Cauchy sequence* if, for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that whenever \(m, n \geq N\) it follows that \(|a_n - a_m| < \epsilon\).
Transcribed Image Text:**Definition:** A sequence \((a_n)\) is called a *Cauchy sequence* if, for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that whenever \(m, n \geq N\) it follows that \(|a_n - a_m| < \epsilon\).
**Problem Statement:** Let \((x_n)\) and \((y_n)\) be Cauchy sequences. Using only the definition of Cauchy (and not using Theorem 2.6.4, or Algebraic Limit Theorems) prove that \((x_n + y_n)\) is Cauchy. --- **Explanation:** This problem involves proving that the sum of two Cauchy sequences is itself a Cauchy sequence. Here, you are restricted to using only the fundamental definition of a Cauchy sequence. A Cauchy sequence is defined as a sequence where, for every positive real number \(\epsilon\), there exists a positive integer \(N\) such that for all integers \(m, n \geq N\), the difference \(|x_m - x_n|\) is less than \(\epsilon\). In this task, you are asked to apply this definition directly (without invoking certain theorems) to show that the sequence \((x_n + y_n)\) satisfies the same property, and therefore is also Cauchy.
Transcribed Image Text:**Problem Statement:** Let \((x_n)\) and \((y_n)\) be Cauchy sequences. Using only the definition of Cauchy (and not using Theorem 2.6.4, or Algebraic Limit Theorems) prove that \((x_n + y_n)\) is Cauchy. --- **Explanation:** This problem involves proving that the sum of two Cauchy sequences is itself a Cauchy sequence. Here, you are restricted to using only the fundamental definition of a Cauchy sequence. A Cauchy sequence is defined as a sequence where, for every positive real number \(\epsilon\), there exists a positive integer \(N\) such that for all integers \(m, n \geq N\), the difference \(|x_m - x_n|\) is less than \(\epsilon\). In this task, you are asked to apply this definition directly (without invoking certain theorems) to show that the sequence \((x_n + y_n)\) satisfies the same property, and therefore is also Cauchy.
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