  I prove that the Bethe roots describing either the ground state or a certain
class of "particle-hole" excited states of the XXZ spin-$1/2$ chain in any
sector with magnetisation $\mathfrak{m} \in [0;1/2]$ exist and form, in the
infinite volume limit, a dense distribution on a subinterval of $\mathbb{R}$.
The results holds for any value of the anisotropy $\Delta \geq -1 $. In fact, I
establish an even stronger result, namely the existence of an all order
asymptotic expansion of the counting function associated with such roots. As a
corollary, these results allow one to prove the existence and form of the
infinite volume limit of various observables attached to the model -the
excitation energy, momentum, the zero temperature correlation functions, so as
to name a few- that were argued earlier in the literature.
