We prove uniform convergence in Lipschitz domains in $\mathbb{R}^n$ (and in $C^{1,1}$ domains in the Heisenberg group) of approximations to $p$-harmonic functions obtained using the natural $p$-means introduced by Ishiwata, Magnanini, and Wadade in 2017. This talk is based in joint work with Andr\'as Domokos and Diego Ricciotti (Sacramento) and Bianca Stroffolini (Naples)