  Consider a charged, perfect quantum gas, in the effective mass approximation,
and in the grand-canonical ensemble. We prove in this paper that the
generalized magnetic susceptibilities admit the thermodynamic limit for all
admissible fugacities, uniformly on compacts included in the analyticity domain
of the grand-canonical pressure.
  The problem and the proof strategy were outlined in \cite{3}. In \cite{4} we
proved in detail the pointwise thermodynamic limit near $z=0$. The present
paper is the last one of this series, and contains the proof of the uniform
bounds on compacts needed in order to apply Vitali's Convergence Theorem.
