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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.

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.

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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part C and D

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.
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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