10.2 Theorem. All bounded monotone sequences converge. Proof Let (sn) be a bounded increasing sequence. Let S denote the set {sn n € N}, and let u = sup S. Since S is bounded, u represents a real number. We show lim sn = u. Let € > 0. Since u — e is not an upper bound for S, there exists N such that sy > u − e. Since (sn) is increasing, we have sã ≤ sʼn for all n ≥ N. Of course, sn ≤ u for all n, so n > N implies u − € < Sn ≤ u, which implies |sn − u| < €. This shows lim sñ = u. The proof for bounded Exercise 10.2. decreasing sequences is left to

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Prove the bounded decreasing sequence 

10.2 Theorem.
All bounded monotone sequences converge.
Proof
e is not an
Let (sn) be a bounded increasing sequence. Let S denote the set
{sn : n € N}, and let u = sup S. Since S is bounded, u represents a
real number. We show lim sn = u. Let € > 0. Since u
upper bound for S, there exists N such that sy > u − €. Since (sn)
is increasing, we have sN ≤ sn for all n ≥ N. Of course, sn ≤ u for
all n, so n > N implies u − € < Sn ≤ u, which implies |sn – u| < e.
This shows lim sñ = u.
decreasing sequences is left to
The proof for bounded
Exercise 10.2.
Transcribed Image Text:10.2 Theorem. All bounded monotone sequences converge. Proof e is not an Let (sn) be a bounded increasing sequence. Let S denote the set {sn : n € N}, and let u = sup S. Since S is bounded, u represents a real number. We show lim sn = u. Let € > 0. Since u upper bound for S, there exists N such that sy > u − €. Since (sn) is increasing, we have sN ≤ sn for all n ≥ N. Of course, sn ≤ u for all n, so n > N implies u − € < Sn ≤ u, which implies |sn – u| < e. This shows lim sñ = u. decreasing sequences is left to The proof for bounded Exercise 10.2.
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