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8. Prove (from the definition of a limit of a sequence) that if the sequence {a} tends to limit L as 11 → 00, then for any fixed number M > 0, the sequence {Ma} tends to the limit ML.

8. Prove (from the definition of a limit of a sequence) that if the sequence {a} tends to limit L as 11 → 00, then for any fixed number M > 0, the sequence {Ma} tends to the limit ML.

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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8. Prove (from the definition of a limit of a sequence) that if the sequence {a} tends to limit L as 11 → 00, then for any fixed number M > 0, the sequence {Ma} tends to the limit ML. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 9. Given an infinite collection An, n=1,2,... of intervals of the real line, their intersection is defined to be N₁4₂₁ = {x|(\n)(x = A)} Give an example of a family of intervals A, n=1,2,..., such that A₂+1 CA for all 1 and 1A=0. Prove that your example has the stated property. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 10. Give an example of a family of intervals An=1,2,..., such that A+1 CA, for all 1 and 4 consists of a single real number. Prove that your example has the stated property.
Transcribed Image Text:8. Prove (from the definition of a limit of a sequence) that if the sequence {a} tends to limit L as 11 → 00, then for any fixed number M > 0, the sequence {Ma} tends to the limit ML. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 9. Given an infinite collection An, n=1,2,... of intervals of the real line, their intersection is defined to be N₁4₂₁ = {x|(\n)(x = A)} Give an example of a family of intervals A, n=1,2,..., such that A₂+1 CA for all 1 and 1A=0. Prove that your example has the stated property. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 10. Give an example of a family of intervals An=1,2,..., such that A+1 CA, for all 1 and 4 consists of a single real number. Prove that your example has the stated property.
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