The space of persistence diagrams on $n$ points coarsely embeds into Hilbert space

We prove that the space of persistence diagrams on $n$ points (with bottleneck or a Wasserstein distance) coarsely embeds into Hilbert by showing it is asymptotic dimension $2n$. Such an embedding enables utilisation techniques diagrams. also when number not bounded, corresponding spaces do have finite dimension. Furthermore, in case distance, does embed space.