Cantor proved that the set of real numbers is uncountable---it cannot be put in bijective correspondence with the natural numbers---but the proof is a simple diagonalization: if the real numbers could be put on a list $z_0$, $z_1$, and so on, then design a real number $d$ whose $n$-th digit difffers from the $n$-th digit of $z_n$. Thus, $d\neq z_n$ for every $n$, contradiction!

So the proof seems trivial, perhaps especially now that diagonalization is (as a result) a standard proof method, but the theorem nevertheless seems profound. It was even controversial for various reasons at the time, and certainly it opened up a completely new understanding and treatment of infinity in mathematics.

Cantor proved that the set of real numbers is uncountable---it cannot be put in bijective correspondence with the natural numbers---but the proof is a simple diagonalization: if the real numbers could be put on a list $z_0$, $z_1$, and so on, then design a real number $d$ whose $n$-th digit differs from the $n$-th digit of $z_n$. Thus, $d\neq z_n$ for every $n$, contradiction!

So the proof seems trivial, perhaps especially now that diagonalization is (as a result) a standard proof method, but the theorem nevertheless seems profound. It was even controversial for various reasons at the time, and certainly it opened up a completely new understanding and treatment of infinity in mathematics.