Asymptotic cones and Assouad-Nagata dimension

We prove the dimension of any asymptotic cone over a metric space X does not exceed the asymptotic Assouad-Nagata dimension of X. This improves a result of Dranishnikov and Smith who showed that dim(Y) does not exceed asymptotic Assouad-Nagata dimension of X for all separable subsets Y of special asymptotic cones of X over an exponential ultrafilter. We also show that Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.