Page 110 - 'Blast_Into_Math

Blast into Math! Prime nummers: indestructimle muilding mlocks

or there is some

z 2 ∈ S 1 \{z 1 }.

Since the set S 1 contains finitely many elements, there is p ∈ N such that

p
S 1 = {z n } n=1 ,

and

if n = m.
z n = z m

We have assigned a unique natural number to the elements of

1

S k = S 1 .
k=1

We can complete the proof by induction. We have proven the base case by assigning a unique natural
number to each element of

1

S k .
k=1

Now we assume we have assigned a unique natural number to the elements of

k

S j ,
j=1

for k> 1. Since each S j is a finite set, there are finitely many elements in the union

k

S j ,
j=1

so there is N ∈ N such that

k

N
{z n } n=1 = S j ,
j=1

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