  Subdiffusive motion takes place at a much slower timescale than diffusive
motion. As a preliminary step to studying reaction-subdiffusion pulled fronts,
we consider here the hyperbolic limit $(t,x) \to (t/\varepsilon,
x/\varepsilon)$ of an age-structured equation describing the subdiffusive
motion of, e.g., some protein inside a biological cell. Solutions of the
rescaled equations are known to satisfy a Hamilton-Jacobi equation in the
formal limit $\varepsilon \to 0$. In this work we derive uniform Lipschitz
estimates, and establish the convergence towards the viscosity solution of the
limiting Hamilton-Jacobi equation. The two main obstacles overcome in this work
are the non-existence of an integrable stationary measure, and the importance
of memory terms in subdiffusion.
