The Real And Complex Number Systems

Proof: Say {ar + b : a ∈ Z, b ∈ Z} = S, and since r ∈ Q c , then by Exercise
1.16, there are infinitely many rational numbers h/k with k > 0 such
that |kr − h| < 1 . Consider (x − δ, x + δ) := I, where δ > 0, and thus choosing
k 0 large enough so that 1/k 0 < δ. Define L = |k 0 r − h 0 | , then we have
k
sL ∈ I for some s ∈ Z. So, sL = (±) [(sk 0 ) r − (sh 0 )] ∈ S. That is, we have
proved that S is dense in R.
1.17 Let x be a positive rational number of the form
x =
n∑
k=1
where each a k is nonnegative integer with a k ≤ k − 1 for k ≥ 2 and a n > 0.
Let [x] denote the largest integer in x. Prove that a 1 = [x] , that a k =
[k!x] − k [(k − 1)!x] for k = 2, ..., n, and that n is the smallest integer such
that n!x is an integer. Conversely, show that every positive rational number
x can be expressed in this form in one and only one way.
and
Proof: (⇒)First,
[ ]
n∑ a k
[x] = a 1 +
k!
k=2
[ n∑
]
a k
= a 1 + since a 1 ∈ N
k!
k=2
n∑ a k
n∑
= a 1 since
k! ≤ k − 1
k!
k=2
k=2
Second, fixed k and consider
k!x = k!
n∑
j=1
(k − 1)!x = (k − 1)!
=
k−1
a j
j! = k! ∑
n∑
j=1
j=1
a k
k! ,
n∑
k=2
1
(k − 1)! − 1 k! = 1 − 1 n! < 1.
a j
j! + a k + k!
k−1
a j
j! = (k − 1)! ∑
j=1
n∑
j=k+1
a j
j!
a j
n∑
j! + (k − 1)!
j=k
a j
j! .
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