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1. Prove or Disprove the following. a. If {an} is Cauchy sequence and bn is bounded, then {anbn} is Cauchy. b. If {a} and {bn} are Cauchy and bn #0 for all n € N, then {n} si Cauchy. bn C..If {an} and {bn} are Cauchy and sn + b₂ > 0 for all n € N, then 1 ■If an + bn cannot converge to zero.

1. Prove or Disprove the following. a. If {an} is Cauchy sequence and bn is bounded, then {anbn} is Cauchy. b. If {a} and {bn} are Cauchy and bn #0 for all n € N, then {n} si Cauchy. bn C..If {an} and {bn} are Cauchy and sn + b₂ > 0 for all n € N, then 1 ■If an + bn cannot converge to zero.

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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1. Prove or Disprove the following. a. If {an} is Cauchy sequence and be is bounded, then {anbn} is Cauchy. an bn b. If {a} and {bn} are Cauchy and bn ‡ 0 for all n € N, then {n} si Cauchy. C..If {an} and {bn} are Cauchy and sn + b₂ > 0 for all n € N, then 1 cannot converge to zero. an + bn 2. Prove that if the sequence {an} satisfies |an| ≤ for all n E N, then {an} is Cauchy. 2n² + 3 n³ +5n² + 3n+1
Transcribed Image Text:1. Prove or Disprove the following. a. If {an} is Cauchy sequence and be is bounded, then {anbn} is Cauchy. an bn b. If {a} and {bn} are Cauchy and bn ‡ 0 for all n € N, then {n} si Cauchy. C..If {an} and {bn} are Cauchy and sn + b₂ > 0 for all n € N, then 1 cannot converge to zero. an + bn 2. Prove that if the sequence {an} satisfies |an| ≤ for all n E N, then {an} is Cauchy. 2n² + 3 n³ +5n² + 3n+1
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