  The purpose of this paper is to prove a uniform convergence rate of the
solutions of the $p$-Laplace equation $\Delta_p u = 0$ with Dirichlet boundary
conditions to the solution of the infinity-Laplace equation $\Delta_\infty u =
0$ as $p\to\infty$. The rate scales like $p^{-1/4}$ for general solutions of
the Dirichlet problem and like $p^{-1/2}$ for solutions with positive gradient.
An explicit example shows that it cannot be better than $p^{-1}$. The proof of
this result solely relies on the comparison principle with the fundamental
solutions of the $p$-Laplace and the infinity-Laplace equation, respectively.
Our argument does not use viscosity solutions, is purely metric, and is
therefore generalizable to more general settings where a comparison principle
with H\"older cones and H\"older regularity is available.
