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The theorem is attached. Thank you.

The theorem is attached. Thank you.

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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The theorem is attached. Thank you.

Let ACR. Prove that A is closed and bounded if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Deduce Theorem 4.12 from this result.
Transcribed Image Text:Let ACR. Prove that A is closed and bounded if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Deduce Theorem 4.12 from this result.
THEOREM 4.12. Let A CR. Then the following are equivalent. (1) A is compact. (2) A is closed and bounded. (3) If (an) is a sequence of numbers in A, then there is a subsequence (an) that converges to a point in A.
Transcribed Image Text:THEOREM 4.12. Let A CR. Then the following are equivalent. (1) A is compact. (2) A is closed and bounded. (3) If (an) is a sequence of numbers in A, then there is a subsequence (an) that converges to a point in A.
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