s1 - Math 5615H: Introduction to Analysis I. Fall 2011...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #1. Problems and short Solutions. #1. Prove that 6 and 2 + 3 are NOT rational. Proof. If p := 6 Q , then p 2 = 6, and we get a contradiction in the same way as in Example 1.1 in the textbook. If q := 2 + 3 Q , then q 2 = 5 + 2 6, and 6 = ( q 2 - 5) / 2 Q , in contradiction with the previous part. #2. Let A and B be bounded sets in R . Consider the algebraic sum of A and B , A + B := { x R : x = a + b for some a A and b B } . Show that sup( A + B ) = sup A + sup B . Proof. Denote α := sup A, β := sup B, γ := sup( A + B ). x A + B, a A and b B such that x = a + b. Then x = a + b α + β , and γ α + β . On the other hand, a A, b B , we have a + b A + B = a + b γ = α + b γ = α + β γ. Therefore, γ = α + β. #3. Find sup A, where A := { x R : x 2 < 3 x - 2 } . Solution. We can write A = (1 , 2). By verifying the properties (i) and (ii) in Definition 1.8,
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