Let A be a nonempty bounded subset of R such that inf A = 3 and...

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Let A be a nonempty bounded subset of R such that inf A = 3 and sup A = 8. Let B = {ry – 18/(r + y) : a E [6, 10) Q, y € A}. Prove that B is bounded. Determine (with proof) the infimum and supremum of B. (b) (5 marks) Determine (with proof) the infimum of the set S = {r :x ERand there exist b, cE [-1,1) such that r + br + c= 0}. (c) (10 marks) Let A1, A2, A3,... [0, 1] such that A1N A2n A3n.. is nonempty. If sup{inf A, : n = 1,2, 3, ...} = inf{sup A, : n = 1,2, 3, ...}, .3B then prove that A n A2 n A3 n... has exactly one element. Let A be a nonempty bounded subset of R such that inf A = 3 and sup A = 8. Let B = {ry – 18/(r + y) : a E [6, 10) Q, y € A}. Prove that B is bounded. Determine (with proof) the infimum and supremum of B. (b) (5 marks) Determine (with proof) the infimum of the set S = {r :x ERand there exist b, cE [-1,1) such that r + br + c= 0}. (c) (10 marks) Let A1, A2, A3,... [0, 1] such that A1N A2n A3n.. is nonempty. If sup{inf A, : n = 1,2, 3, ...} = inf{sup A, : n = 1,2, 3, ...}, .3B then prove that A n A2 n A3 n... has exactly one element.