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Take any list of reals - for example, the algebraics: \(\{a_i\}\).
Cover each \(a_i\) with an interval of size \(2^{-i}\).
Total length is 1, but every number in the list is covered.
So every list is incomplete, hence the reals are uncountable.

Corollary: The algebraics are countable, hence there are transcendentals.

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