12. For each n € N, define An = [0,1 + ½] and B₂ = [0, 1 − 1]. - (a) Prove that Uº₁ An is an interval, and describe this interval explicitly. n=1 (b) Prove that 1 An is an interval, and describe this interval explicitly. =1 n=1 (c) Prove that U-₁ B₁ is an interval, and describe this interval explicitly. (d) Prove that ₁ B₂ is an interval, and describe this interval explicitly. n=1
12. For each n € N, define An = [0,1 + ½] and B₂ = [0, 1 − 1]. - (a) Prove that Uº₁ An is an interval, and describe this interval explicitly. n=1 (b) Prove that 1 An is an interval, and describe this interval explicitly. =1 n=1 (c) Prove that U-₁ B₁ is an interval, and describe this interval explicitly. (d) Prove that ₁ B₂ is an interval, and describe this interval explicitly. n=1
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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![**Problem 12.** For each \( n \in \mathbb{N} \), define \( A_n = \left[0, 1 + \frac{1}{n}\right] \) and \( B_n = \left[0, 1 - \frac{1}{n}\right] \).
**(a)** Prove that \( \bigcup_{n=1}^{\infty} A_n \) is an interval, and describe this interval explicitly.
**(b)** Prove that \( \bigcap_{n=1}^{\infty} A_n \) is an interval, and describe this interval explicitly.
**(c)** Prove that \( \bigcup_{n=1}^{\infty} B_n \) is an interval, and describe this interval explicitly.
**(d)** Prove that \( \bigcap_{n=1}^{\infty} B_n \) is an interval, and describe this interval explicitly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F912a24df-6035-40a5-a35d-878c2e7f673f%2F8f992900-9c9b-4053-ae07-cb1055c745b6%2F9uo5an_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 12.** For each \( n \in \mathbb{N} \), define \( A_n = \left[0, 1 + \frac{1}{n}\right] \) and \( B_n = \left[0, 1 - \frac{1}{n}\right] \).
**(a)** Prove that \( \bigcup_{n=1}^{\infty} A_n \) is an interval, and describe this interval explicitly.
**(b)** Prove that \( \bigcap_{n=1}^{\infty} A_n \) is an interval, and describe this interval explicitly.
**(c)** Prove that \( \bigcup_{n=1}^{\infty} B_n \) is an interval, and describe this interval explicitly.
**(d)** Prove that \( \bigcap_{n=1}^{\infty} B_n \) is an interval, and describe this interval explicitly.
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