Let A and B be subsets of R which are bounded above. Show that sup (AnB) ≤ min {sup (A), sup (B)} Give an example to show that inequality could be strict. Hint: Let x An B, then x EA and x E B. • Explain why ● Explain why can you conclude that ● ● ● ● x≤ sup (A) and x ≤ sup (B). x ≤ min {sup (A), sup (B)}, for all x EAN B Observe that the above shows that min {sup (A), sup (B)} is an upper bound for AnB. Now, observe that sup (An B) is the least upper bound for An B. Explain why you can conclude that sup (An B) ≤ min {sup (A), sup (B)} Let A = [0, 1) and B = [1, then U Find sup (An B), sup (A), and sup B Conclude that sup (An B) < min {sup (A), sup (B)
Let A and B be subsets of R which are bounded above. Show that sup (AnB) ≤ min {sup (A), sup (B)} Give an example to show that inequality could be strict. Hint: Let x An B, then x EA and x E B. • Explain why ● Explain why can you conclude that ● ● ● ● x≤ sup (A) and x ≤ sup (B). x ≤ min {sup (A), sup (B)}, for all x EAN B Observe that the above shows that min {sup (A), sup (B)} is an upper bound for AnB. Now, observe that sup (An B) is the least upper bound for An B. Explain why you can conclude that sup (An B) ≤ min {sup (A), sup (B)} Let A = [0, 1) and B = [1, then U Find sup (An B), sup (A), and sup B Conclude that sup (An B) < min {sup (A), sup (B)
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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Transcribed Image Text:Let A and B be subsets of R which are bounded above. Show that
sup (AnB) ≤ min {sup (A), sup (B)}
Give an example to show that inequality could be strict.
Hint: Let x An B, then x EA and x E B.
●
• Explain why
●
Explain why can you conclude that
●
●
●
●
x≤ sup (A) and x ≤ sup (B).
x ≤ min {sup (A), sup (B)}, for all x E An B
Observe that the above shows that min {sup (A), sup (B)} is an upper bound for
AnB.
Now, observe that sup (An B) is the least upper bound for An B. Explain why
you can conclude that
sup (An B) ≤ min {sup (A), sup (B)}
Let A = [0, 1) and B = [1, then
U
Find sup (An B), sup (A), and sup B
Conclude that sup (An B) < min {sup (A), sup (B)
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