On the elementary symmetric functions of $1, 1/2, \ldots , 1/n$

In 1946, P. Erd\H os and I. Niven proved that there are only finitely many positive integers $n$ for which one or more elementary symmetric functions of $1, 1/2, \ldots , 1/n$ are integers. In this paper we solve this old problem by showing that if $n\ge 4$, then none of elementary symmetric functions of $1, 1/2, \ldots , 1/n$ is an integer.