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.