A Proof Of Cantor's Theorem

We present a short proof of Cantor's Theorem (circa 1870s): if $a_n \cos nx + b_n \sin nx \to 0$ for each $x$ in some (nonempty) open interval, where $a_n, b_n$ are sequences of complex numbers, then $a_n$ and $b_n$ converge to 0.