Page 108 - 'Blast_Into_Math

Blast into Math! Prime nummers: indestructimle muilding mlocks

The following proposition tells us an interesting fact about infinity:

If a finite part is removed from infinity, what remains is still infinity.

Proposition 5.4.3 (Infinite Proposition). Let S be a set which contains infinitely many elements. If Y ⊆ S
is a set which contains finitely many elements, then the set S \ Y also contains infinitely many elements.

Proof: By the definition of an infinite set, for each n ∈ N , S contains at least n elements. If Y = ∅ ,
then S \ Y = S contains infinitely many elements. Otherwise, there is k ∈ N such that Y contains
k elements. By the definition of infinite set, since n + k ∈ N for each n ∈ N , S contains at least

n + k elements, and since Y contains precisely k elements, this means that S \ Y contains at least
n elements, for each n ∈ N. The set S \ Y fits perfectly into the definition of an infinite set.

♥

The next theorem shows that we can zip finite sets together into one countable set.

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