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Department of Mathematics, University of Toronto
Problem Set Supplement #1
MAT 137Y, 200809 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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