2. In the following, let SCR be non-empty and bounded above. In this problem, we will see an interesting connection between the supremum and infinite sets. (a) Show that there exists a sequence {n} with an € S for all n € N such that lim n = sup S. n→∞ (Hint: Look at Q8 on HW1 and HW2) (b) Suppose sup SS. Show that there exists a strictly monotone increasing se- quence {n} with yn ES for all n € N. (Hint: You may want to approach this problem inductively. Take some y₁ € S, and note that sup S − y₁ > 0 (why?). Use this with HW1 Q8 to find y2, etc... When doing induction, you may want to prove that sup S - yp> 0 for all p = N.) (c) Suppose sup S & S. Show that there exists a countably infinite subset EC S. (Hint: Consider the set {yn: n € N} defined for the sequence {yn} in (b). Can you find some bijection between this set and N?)
2. In the following, let SCR be non-empty and bounded above. In this problem, we will see an interesting connection between the supremum and infinite sets. (a) Show that there exists a sequence {n} with an € S for all n € N such that lim n = sup S. n→∞ (Hint: Look at Q8 on HW1 and HW2) (b) Suppose sup SS. Show that there exists a strictly monotone increasing se- quence {n} with yn ES for all n € N. (Hint: You may want to approach this problem inductively. Take some y₁ € S, and note that sup S − y₁ > 0 (why?). Use this with HW1 Q8 to find y2, etc... When doing induction, you may want to prove that sup S - yp> 0 for all p = N.) (c) Suppose sup S & S. Show that there exists a countably infinite subset EC S. (Hint: Consider the set {yn: n € N} defined for the sequence {yn} in (b). Can you find some bijection between this set and 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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![2. In the following, let \( S \subset \mathbb{R} \) be non-empty and bounded above. In this problem, we will see an interesting connection between the supremum and infinite sets.
(a) Show that there exists a sequence \(\{x_n\}\) with \(x_n \in S\) for all \(n \in \mathbb{N}\) such that
\[
\lim_{n \to \infty} x_n = \sup S.
\]
*(Hint: Look at Q8 on HW1 and HW2)*
(b) Suppose \(\sup S \notin S\). Show that there exists a strictly monotone increasing sequence \(\{y_n\}\) with \(y_n \in S\) for all \(n \in \mathbb{N}\).
*(Hint: You may want to approach this problem inductively. Take some \(y_1 \in S\), and note that \(\sup S - y_1 > 0\) (why?). Use this with HW1 Q8 to find \(y_2\), etc... When doing induction, you may want to prove that \(\sup S - y_p > 0\) for all \(p \in \mathbb{N}\).)*
(c) Suppose \(\sup S \notin S\). Show that there exists a countably infinite subset \(E \subset S\).
*(Hint: Consider the set \(\{y_n : n \in \mathbb{N}\}\) defined for the sequence \(\{y_n\}\) in (b). Can you find some bijection between this set and \(\mathbb{N}\)?)*
(d) Suppose \(\sup S \in S\). Will there always be a countably infinite subset \(E \subset S\)? Either prove the statement, or give a counterexample.
*(Note: finite sets are never countably infinite.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F6aa45169-cd77-4c58-80d4-054764175577%2F0aqy72l_processed.png&w=3840&q=75)
Transcribed Image Text:2. In the following, let \( S \subset \mathbb{R} \) be non-empty and bounded above. In this problem, we will see an interesting connection between the supremum and infinite sets.
(a) Show that there exists a sequence \(\{x_n\}\) with \(x_n \in S\) for all \(n \in \mathbb{N}\) such that
\[
\lim_{n \to \infty} x_n = \sup S.
\]
*(Hint: Look at Q8 on HW1 and HW2)*
(b) Suppose \(\sup S \notin S\). Show that there exists a strictly monotone increasing sequence \(\{y_n\}\) with \(y_n \in S\) for all \(n \in \mathbb{N}\).
*(Hint: You may want to approach this problem inductively. Take some \(y_1 \in S\), and note that \(\sup S - y_1 > 0\) (why?). Use this with HW1 Q8 to find \(y_2\), etc... When doing induction, you may want to prove that \(\sup S - y_p > 0\) for all \(p \in \mathbb{N}\).)*
(c) Suppose \(\sup S \notin S\). Show that there exists a countably infinite subset \(E \subset S\).
*(Hint: Consider the set \(\{y_n : n \in \mathbb{N}\}\) defined for the sequence \(\{y_n\}\) in (b). Can you find some bijection between this set and \(\mathbb{N}\)?)*
(d) Suppose \(\sup S \in S\). Will there always be a countably infinite subset \(E \subset S\)? Either prove the statement, or give a counterexample.
*(Note: finite sets are never countably infinite.)*
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