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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
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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Could you explain how to show 6.17 in detail?

The next theorem completely characterizes the sets in Rstd 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 R' is bounded if and only if there is a number M such that A C [-M, M]. Theorem 6.15 (Heine-Borel Theorem). Let A be a subset of Rstd- 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 E C such that for any x E C, x < m.
Transcribed Image Text:The next theorem completely characterizes the sets in Rstd 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 R' is bounded if and only if there is a number M such that A C [-M, M]. Theorem 6.15 (Heine-Borel Theorem). Let A be a subset of Rstd- 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 E C such that for any x E C, x < m.
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