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)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.6: Inequalities

Problem 80E

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