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Cantor established these results using two constructions.
His first construction shows how to write the real algebraic numbers as a sequence a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ....
In other words, the real algebraic numbers are countable.
Cantor starts his second construction with any sequence of real numbers.
Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence.
Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence — that is, the real numbers are not countable.
By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number.
Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.
Cantor's next article contains a construction that proves the set of transcendental numbers has the same " power " ( see below ) as the set of real numbers .< ref > Cantor's construction starts with the set of transcendentals T and removes a countable subset

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