8. a) The Monotonic Convergence Theorem (MCT) states that if a sequence of real numbers {an} satisfies an < an+1 < M Vn e N, for some fixed value of the constant M, then such sequence converges to a real number. Sketch the proof of the MCT and give 2 examples of sequences that satisfy the hypotheses of the theorem, and two examples of sequences that do not satisfy all of the hypotheses;
8. a) The Monotonic Convergence Theorem (MCT) states that if a sequence of real numbers {an} satisfies an < an+1 < M Vn e N, for some fixed value of the constant M, then such sequence converges to a real number. Sketch the proof of the MCT and give 2 examples of sequences that satisfy the hypotheses of the theorem, and two examples of sequences that do not satisfy all of the hypotheses;
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![8. a) The Monotonic Convergence Theorem (MCT) states that if a sequence
of real numbers {an} satisfies
An < an+1 < M
Vn E N,
for some fixed value of the constant M, then such sequence converges to a
real number. Sketch the proof of the MCT and give 2 examples of sequences
that satisfy the hypotheses of the theorem, and two examples of sequences
that do not satisfy all of the hypotheses;
b) state and sketch the proof of the Nested Interval Theorem (NIT);
c) what is the most crucial ingredient of the proofs of the MCT and the NIT
above? Please, illustrate an algebraic ordered field and a counterexample for
which the above two theorems do not hold;
d) do the intervals in the statement of the Nested Interval Theorem need to
contain their endpoints for the theorem to hold? Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a7d9fc5-182b-4619-bcf5-f19f3bccad39%2F3915d01c-9c08-4652-8db4-0630319d41ff%2F0il4vo7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8. a) The Monotonic Convergence Theorem (MCT) states that if a sequence
of real numbers {an} satisfies
An < an+1 < M
Vn E N,
for some fixed value of the constant M, then such sequence converges to a
real number. Sketch the proof of the MCT and give 2 examples of sequences
that satisfy the hypotheses of the theorem, and two examples of sequences
that do not satisfy all of the hypotheses;
b) state and sketch the proof of the Nested Interval Theorem (NIT);
c) what is the most crucial ingredient of the proofs of the MCT and the NIT
above? Please, illustrate an algebraic ordered field and a counterexample for
which the above two theorems do not hold;
d) do the intervals in the statement of the Nested Interval Theorem need to
contain their endpoints for the theorem to hold? Explain your answer.
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