Vertical Ends of Constant Mean Curvature H=1/2 in H^2\times R

We prove a vertical halfspace theorem for surfaces with constant mean curvature $H=1\mathrm{/}2,$ properly immersed in the product space ${\text{h}}^2\times \text{re},$ where ${\text{h}}^2$ is the hyperbolic plane and $\text{re}$ is the set of real numbers. The proof is a geometric application of the classical maximum principle for second order elliptic PDE, using the family of non compact rotational $H=1\mathrm{/}2$ surfaces in ${\text{h}}^2\times \text{re}\mathrm{.}$