(Sequences (an).) We say a sequence (an) is eventually in a set A if there exists a natural number NEN such that an EA for all natural numbers n ≥ N. Let (an) be a sequence of positive numbers an> 0 such that (an) is eventually in the closed interval A = [0, 2]. Follow the steps below to show that the sequence (an) is a bounded sequence. (a) Let M₁ 2. Show that there is a natural number NEN such that for all n > N. |an| ≤ M₁ (b) Let M₂ = max{|a₁|, |a₂|,..., |aN-1}, where N is the same as in part (a). Explain why |an| ≤ M₂ for all n {1, 2, ..., N-1}. (c) Let M = max{M₁, M₂}. Explain why |an| ≤ M for all n € N.
(Sequences (an).) We say a sequence (an) is eventually in a set A if there exists a natural number NEN such that an EA for all natural numbers n ≥ N. Let (an) be a sequence of positive numbers an> 0 such that (an) is eventually in the closed interval A = [0, 2]. Follow the steps below to show that the sequence (an) is a bounded sequence. (a) Let M₁ 2. Show that there is a natural number NEN such that for all n > N. |an| ≤ M₁ (b) Let M₂ = max{|a₁|, |a₂|,..., |aN-1}, where N is the same as in part (a). Explain why |an| ≤ M₂ for all n {1, 2, ..., N-1}. (c) Let M = max{M₁, M₂}. Explain why |an| ≤ M for all n € N.
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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Question
![(Sequences (an).) We say a sequence (an) is eventually in a set A if there exists a natural
number NEN such that an € A for all natural numbers n ≥ N. Let (an) be a sequence of
positive numbers an> 0 such that (an) is eventually in the closed interval A = [0, 2]. Follow
the steps below to show that the sequence (an) is a bounded sequence.
(a) Let M₁ = 2. Show that there is a natural number N € N such that
for all n ≥ N.
|an| ≤ M₁
(b) Let M₂ = max{|a₁|, |a₂|, . . . , |ax-1|}, where N is the same as in part (a). Explain why
for all n € {1,2, ..., N – 1}.
|an| ≤ M₂
(c) Let M
=
max{M₁, M₂}. Explain why
|an| ≤ M
for all n E N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6f2d17b-82dd-4cad-827a-23a867087cf6%2F308fa3b7-6c5c-4b34-8a90-e93d9f653f5c%2Fhhyrtms_processed.png&w=3840&q=75)
Transcribed Image Text:(Sequences (an).) We say a sequence (an) is eventually in a set A if there exists a natural
number NEN such that an € A for all natural numbers n ≥ N. Let (an) be a sequence of
positive numbers an> 0 such that (an) is eventually in the closed interval A = [0, 2]. Follow
the steps below to show that the sequence (an) is a bounded sequence.
(a) Let M₁ = 2. Show that there is a natural number N € N such that
for all n ≥ N.
|an| ≤ M₁
(b) Let M₂ = max{|a₁|, |a₂|, . . . , |ax-1|}, where N is the same as in part (a). Explain why
for all n € {1,2, ..., N – 1}.
|an| ≤ M₂
(c) Let M
=
max{M₁, M₂}. Explain why
|an| ≤ M
for all n E N.
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