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Blast into Math! Prime nummers: indestructimle muilding mlocks

contains infinitely many elements. Therefore there must be some j> k +1 such that

N
S j \{z n } n=1 = ∅.

N = ∅ . To do this we can use the set
Let’s look for the smallest j> k +1 such that S j \{z n }
n=1
N
{j ∈ N such that j ≥ k +1 and S j \{z n } n=1 = ∅}.

This is a non-empty set of integers which is bounded below by k +1, so by the GLB Property it contains
a unique smallest element. Let’s call it g . Then

N
S g \{z n } n=1 = ∅,
so there is m ∈ N such that

N m
S g \{z n } n=1 = {s n } n=1 .

Then we define

z N+n = s n , for n =1,...,m.

This sets the induction escalator into motion, proving the theorem.

♥

We can now use the Zipper Theorem to prove that the set of all rational numbers is countable.

Theorem 5.4.5 The set of all rational numbers is countable.

Proof: To use the Zipper Theorem we need to group the rational numbers into finite sets S k . We can

do this with help from the Rational Theorem. Each rational number can be written as a quotient

p
,
q

with p ∈ Z and q ∈ N . Let’s think about |p| + q . Since p ∈ Z and q ∈ N ,

|p| + q ≥ 1.

For which rational number is |p| + q =1? Since q ∈ N , q ≥ 1, so the only way that

|p| + q =1

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