Department of Mathematics, University of Toronto Problem Set Supplement #1 MAT 137Y, 2008-09 Winter Session Not to be handed in. Assignment Posted/Revised: November 4, 2008, 11:03 While the following problems do not have to be handed in, you will be responsible for this material for the ﬁrst term test. 1. SHE 11.1: 1, 3, 5, 7, 11, 13, 21, 23. 2. One property of rational and irrational numbers we have used in our problem sets is that the set of rational and irrational numbers are dense ; that is, between any two numbers there exist rational and irrational numbers. Let x , y ∈ R , r , s ∈ Q , and t 6∈ Q . (a) Suppose x and y are real numbers such that y-x > 1. Prove that there exists k ∈ Z such that x < k < y . Hint: Let l be the largest integer which satisﬁes l ≤ x . Consider the integer l + 1. (b) Given any two numbers x and y such that x < y , prove that there exists an integer n such that y-x > 1 n . (c) Using part (b) and then part (a), show that there exists a rational number
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