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Let an be a bounded sequence of real numbers. Let s = sup{an: n ≥ 1} {ann € N}. Show that there is a subsequence of an that and s increases to s.

Let an be a bounded sequence of real numbers. Let s = sup{an: n ≥ 1} {ann € N}. Show that there is a subsequence of an that and s increases to s.

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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Let an be a bounded sequence of real numbers. Let s = sup {an : n ≥ 1} and s ∈/ {an : n ∈ N}. Show that there is a subsequence of an that increases to s.
**Problem Statement** Let \( a_n \) be a bounded sequence of real numbers. Let \( s = \sup \{ a_n : n \geq 1 \} \) and \( s \notin \{ a_n : n \in \mathbb{N} \} \). Show that there is a subsequence of \( a_n \) that increases to \( s \). **Explanation** - **\( a_n \):** A sequence of real numbers that is bounded. - **\( s = \sup \{ a_n : n \geq 1 \} \):** The supremum (least upper bound) of the sequence. - **\( s \notin \{ a_n : n \in \mathbb{N} \} \):** The value \( s \) is not an element of the sequence. **Objective** Demonstrate that there exists a subsequence of \( a_n \) which converges to \( s \). This type of problem often arises in the context of real analysis, focusing on properties of sequences and their limits. The goal is to use the definition of supremum and subsequences to highlight the existence of such a convergent subsequence.
Transcribed Image Text:**Problem Statement** Let \( a_n \) be a bounded sequence of real numbers. Let \( s = \sup \{ a_n : n \geq 1 \} \) and \( s \notin \{ a_n : n \in \mathbb{N} \} \). Show that there is a subsequence of \( a_n \) that increases to \( s \). **Explanation** - **\( a_n \):** A sequence of real numbers that is bounded. - **\( s = \sup \{ a_n : n \geq 1 \} \):** The supremum (least upper bound) of the sequence. - **\( s \notin \{ a_n : n \in \mathbb{N} \} \):** The value \( s \) is not an element of the sequence. **Objective** Demonstrate that there exists a subsequence of \( a_n \) which converges to \( s \). This type of problem often arises in the context of real analysis, focusing on properties of sequences and their limits. The goal is to use the definition of supremum and subsequences to highlight the existence of such a convergent subsequence.
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