  The paper is devoted to the large scale geometry of the Heisenberg group
$\mathbb H$ equipped with left-invariant Riemannian distances.
  We prove that two such distances have bounded difference if and only if they
are asymptotic, i.e., their ratio goes to one, at infinity.
  Moreover, we show that for every left-invariant Riemannian distance $d$ on
$\mathbb H$ there is a unique subRiemanniann metric $d'$ for which $d-d'$ goes
to zero at infinity, and we estimate the rate of convergence.
  As a first immediate consequence we get that the Riemannian Heisenberg group
is at bounded distance from its asymptotic cone.
  The second consequence, which was our aim, is the explicit description of the
horoboundary of the Riemannian Heisenberg group.
