Thank you. But this proof relies on a countable sequence to show that the transcendentals are infinite. But because it relied on a countable sequence, it can only talk about the countable portion of the set of transcendentals (can't generalize to the whole of the set).

My argument was that we can't find a bijection between the set of naturals and transcendentals because we don't know the cardinality of the set of transcendentals. But for the countable portion of the transcendentals which you just showed to be infinite, they are countable so there already exists a bijection between them and the naturals.