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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