I'm not sure whether my solution for the existence of a minimum on the generalised Rayleigh quotient is. The problem is to show that $$\min\frac{(x,Hx)}{(x,Mx)}$$ has a solution, where $H,M$ are both self adjoint, and $M$ is positive i.e. $(x,Mx)>0\forall x$. I've followed Lax's proof for the Rayleigh quotient pretty closely, by noting that the generalised Rayleigh quotient is homogeneous of degree zero, so we can scale all the vectors onto the unit sphere. Since they are all bounded on the Rayleigh sphere, by Cauchy's thm any sequence of vectors has a convergent subsequence, and so a minimum exists.
I'm not sure whether my explanations are correct, since I've had to interpret the parts that Lax has left out. Is there anything wrong with the proof (e.g. not general enough, flat out wrong, etc)?