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pact set of R, prove that max E

pact set of R, prove that max E

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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**Theorem: Existence of Maximum and Minimum in Compact Sets** *Problem Statement:* Let \( E \) be a compact set of \( \mathbb{R} \). Prove that \(\max E\) and \(\min E\) exist. *Explanation:* In this problem, we consider a set \( E \) that is both closed and bounded, properties which define compactness in the real number system. The task is to demonstrate that within such a set, there are elements which serve as the maximum and minimum values of the set. *Steps to Prove:* 1. **Closed and Bounded:** - Since \( E \) is closed, it includes all its limit points. - Being bounded implies that there exists a real number \( M \) such that every element \( x \in E \) satisfies \(-M \leq x \leq M\). 2. **Existence of Supremum and Infimum:** - By the properties of the real numbers, any non-empty set that is bounded above has a supremum (least upper bound). - Similarly, any non-empty set that is bounded below has an infimum (greatest lower bound). 3. **Containment of Supremum and Infimum in \( E \):** - For a compact set in \( \mathbb{R} \), both the supremum and infimum must actually belong to the set \( E \). This follows since supremum and infimum reached by compact sets are limits, and \( E \) being closed contains these limits. 4. **Conclusion:** - This guarantees that the maximum and minimum elements exist: \(\max E = \sup E\) and \(\min E = \inf E\), both of which lie within \( E \). Through these steps, we have demonstrated that any compact subset of the real numbers possesses both a maximum and a minimum value. These properties are pivotal in the study of real analysis and have numerous applications in calculus and optimization problems.
Transcribed Image Text:**Theorem: Existence of Maximum and Minimum in Compact Sets** *Problem Statement:* Let \( E \) be a compact set of \( \mathbb{R} \). Prove that \(\max E\) and \(\min E\) exist. *Explanation:* In this problem, we consider a set \( E \) that is both closed and bounded, properties which define compactness in the real number system. The task is to demonstrate that within such a set, there are elements which serve as the maximum and minimum values of the set. *Steps to Prove:* 1. **Closed and Bounded:** - Since \( E \) is closed, it includes all its limit points. - Being bounded implies that there exists a real number \( M \) such that every element \( x \in E \) satisfies \(-M \leq x \leq M\). 2. **Existence of Supremum and Infimum:** - By the properties of the real numbers, any non-empty set that is bounded above has a supremum (least upper bound). - Similarly, any non-empty set that is bounded below has an infimum (greatest lower bound). 3. **Containment of Supremum and Infimum in \( E \):** - For a compact set in \( \mathbb{R} \), both the supremum and infimum must actually belong to the set \( E \). This follows since supremum and infimum reached by compact sets are limits, and \( E \) being closed contains these limits. 4. **Conclusion:** - This guarantees that the maximum and minimum elements exist: \(\max E = \sup E\) and \(\min E = \inf E\), both of which lie within \( E \). Through these steps, we have demonstrated that any compact subset of the real numbers possesses both a maximum and a minimum value. These properties are pivotal in the study of real analysis and have numerous applications in calculus and optimization problems.
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