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We prove that the uniform infinite half-plane quadrangulation (UIHPQ), witheither general or simple boundary, equipped with its graph distance, itsnatural area measure, and the curve which traces its boundary, converges in thescaling limit to the Brownian half-plane. The topology of convergence is givenby the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric oncurve-decorated metric measure spaces, which is a generalization of theGromov-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 embeddedinto a common metric space in such a way that the spaces $X_1$ and $X_2$ areclose in the Hausdorff distance, the measures $\mu_1$ and $\mu_2$ are close inthe Prokhorov distance, and the curves $\eta_1$ and $\eta_2$ are close in theuniform distance.

Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology