Q:

More transcendental numbers than natural numbers

Are there any simple proofs that obtain this result? I haven't been able to find one online.

A:

The word "simple" is very tricky. The usual proofs goes like this:

Show that the union, and product, of two countable sets is countable.
Show that $\Bbb Q$ is countable.
Show that $\Bbb Q[x]$ is countable.
Show that The algebraic numbers are countable.
Show that the real numbers are uncountable.
Show that the set of transcendental real numbers is uncountable.

The steps are not very difficult if you have seen them before. But if you haven't seen any set theoretic arguments before hand, then it might not be so simple. 

