For each of the following state- ments about suprema and infima, decide whether it is true or false. If it is true, prove it. If it is false, give an example which exhibits that it is false. (a) If A and B are nonempty and bounded, and ACB, then sup A ≤ sup B. (b) If sup A < inf B for sets A and B (so that A and B are nonempty, A is bounded above and B is bounded below) then there exists c ER so that for all a E A and all be B we have a < c < b. (c) Let A and B be nonempty sets of real numbers and suppose that A is bounded above and B is bounded below. Suppose that there exists CER so that for all a E A and all b E B we have a < c < b. Then Sup A inf B.
For each of the following state- ments about suprema and infima, decide whether it is true or false. If it is true, prove it. If it is false, give an example which exhibits that it is false. (a) If A and B are nonempty and bounded, and ACB, then sup A ≤ sup B. (b) If sup A < inf B for sets A and B (so that A and B are nonempty, A is bounded above and B is bounded below) then there exists c ER so that for all a E A and all be B we have a < c < b. (c) Let A and B be nonempty sets of real numbers and suppose that A is bounded above and B is bounded below. Suppose that there exists CER so that for all a E A and all b E B we have a < c < b. Then Sup A inf B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:For each of the following state-
ments about suprema and infima, decide whether it is true or false. If it is
true, prove it. If it is false, give an example which exhibits that it is false.
(a) If A and B are nonempty and bounded, and ACB, then sup A ≤
sup B.
(b) If sup A < inf B for sets A and B (so that A and B are nonempty, A
is bounded above and B is bounded below) then there exists c ER
so that for all a E A and all be B we have a <c<b.
(c) Let A and B be nonempty sets of real numbers and suppose that A
is bounded above and B is bounded below. Suppose that there exists
CER so that for all a E A and all be B we have a < c < b. Then
sup A inf B.
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