Average Entropy of a Quantum Subsystem

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≤n is ${S}m,n{{0ex}{0ex}}={{0ex}{0ex}}({Σ}{k{{0ex}{0ex}}={{0ex}{0ex}}n+1}ᵐⁿ1/k){-}(m{-}1)/2n$. In this Letter a simple proof of this conjecture is given.