Elementary proof of Brauer's and Nesbitt's theorem on zeros of characters of finite groups

The following has been proven by Brauer and Nesbitt. Let $G$ be a finite group, and let $p$ be a prime. Assume $\chi$ is an irreducible complex character of $G$ such that the order of a $p$-Sylow subgroup of $G$ divides the degree of $\chi$. Then $\chi$ vanishes on all those elements of $G$ whose order is divisible by $p$. The two only known proofs of this theorem use profound methods of representation theory, namely the theory of modular representations or Brauer's characterization of generalized characters. The purpose of this paper is to present a more elementary proof.