  We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with
either general or simple boundary, equipped with its graph distance, its
natural area measure, and the curve which traces its boundary, converges in the
scaling limit to the Brownian half-plane. The topology of convergence is given
by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on
curve-decorated metric measure spaces, which is a generalization of the
Gromov-Hausdorff metric whereby two such spaces $(X_1, d_1 , \mu_1,\eta_1)$ and
$(X_2, d_2 , \mu_2,\eta_2)$ are close if they can be isometrically embedded
into a common metric space in such a way that the spaces $X_1$ and $X_2$ are
close in the Hausdorff distance, the measures $\mu_1$ and $\mu_2$ are close in
the Prokhorov distance, and the curves $\eta_1$ and $\eta_2$ are close in the
uniform distance.
