  It was recently conjectured by D. Page that if a quantum system of Hilbert
space dimension $nm$ is in a random pure state then the average entropy of a
subsystem of dimension $m$ where $m \leq n$ is $ S_{mn} =
\sum^{mn}_{k=n+1}(1/k) - (m-1)/2n$. In this letter this conjecture is proved.
