Average Entropy of a Quantum Subsystem

It was recently conjectured by D. Page that if a quantum system of Hilbert space dimension $\mathrm{nm}$ is in a random pure state then the average entropy of a subsystem of dimension $m$ where $m\le n$ is $S_{m,n}=\left(\Sigma {k=n+1\mathrm{}}^{\mathrm{mn}}1/k\right)-(m-1\mathrm{})/2n$. In this Letter a simple proof of this conjecture is given.