Locally uniform limit of harmonic functions

The locally uniform convergence limit of real harmonic functions on a planar domain is harmonic. On a smaller circle, use the Schwarz integral on the unit disk, rescaled to that radius, to construct holomorphic functions whose real parts are the given functions. Uniform convergence of boundary values gives locally uniform convergence of these integrals. Their limit is holomorphic by the Weierstrass convergence theorem, and its real part is the original limit. Applying this construction locally at every point proves the claim without assuming derivative convergence.