A short proof of Gromov's filling inequality

We give a very short and rather elementary proof of Gromov's filling volume inequality for $n$-dimensional Lipschitz cycles (with integer and $\mathbb{Z}_2$-coefficients) in $L^\infty$-spaces. This inequality is used in the proof of Gromov's systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the difficult step therein.