  K. Mahler introduced the concept of perfect systems in the general theory he
developed for the simultaneous Hermite-Pade approximation of analytic
functions. We prove that Nikishin systems are perfect providing, by far, the
largest class of systems of functions for which this important property holds.
As consequences, in the context of Nikishin systems, we obtain: an extension of
Markov's theorem to simultaneous Hermite-Pade approximation, a general result
on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the
logarithmic asymptotics of general sequences of multiple orthogonal
polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio
asymptotics of mixed type multiple orthogonal polynomials.
