Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C such that for aпу х € С, х < т.
Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C such that for aпу х € С, х < т.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you explain how to show 6.17 in detail?
![The next theorem completely characterizes the sets in ℝ_std that are compact. This theorem, known as the Heine-Borel Theorem, is one of the fundamental theorems about the topology of the line.
Recall a set \( A \) in ℝ¹ is bounded if and only if there is a number \( M \) such that \( A \subseteq [-M, M] \).
**Theorem 6.15 (Heine-Borel Theorem).** Let \( A \) be a subset of ℝ_std. Then \( A \) is compact if and only if \( A \) is closed and bounded.
**Theorem 6.17.** Every compact subset \( C \) of ℝ contains a maximum in the set \( C \), i.e., there is an \( m \in C \) such that for any \( x \in C, \, x \leq m \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56be79ad-be6a-45f1-b897-58d23fd7e62d%2Fbb2c4484-a675-456a-b71c-aa45e1fcf57d%2Frr4xgd_processed.png&w=3840&q=75)
Transcribed Image Text:The next theorem completely characterizes the sets in ℝ_std that are compact. This theorem, known as the Heine-Borel Theorem, is one of the fundamental theorems about the topology of the line.
Recall a set \( A \) in ℝ¹ is bounded if and only if there is a number \( M \) such that \( A \subseteq [-M, M] \).
**Theorem 6.15 (Heine-Borel Theorem).** Let \( A \) be a subset of ℝ_std. Then \( A \) is compact if and only if \( A \) is closed and bounded.
**Theorem 6.17.** Every compact subset \( C \) of ℝ contains a maximum in the set \( C \), i.e., there is an \( m \in C \) such that for any \( x \in C, \, x \leq m \).
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