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For $x,z\in B_{\rho }(y)$, the open balls satisfy $B_{\rho }(x)\subset B_{3\rho }(z)\subset B_{4\rho }(y)$. Nonnegativity and the mean value property for harmonic functions giveTaking the supremum over $x$ and the infimum over $z$ proves the Harnack inequality for harmonic functions:This includes a zero infimum, which forces $u=0$ on the smaller ball.

Shift $u$ by a constant and, if necessary, change sign to obtain an entire nonnegative harmonic function $v$. The Harnack inequality for harmonic functions on arbitrarily large balls centred at $x_0$ gives $v(x)\le 3^{n}v(x_0)$ for every $x$. Thus $v$ is globally bounded. The interior derivative estimate for a harmonic function now gives $|Dv(x)|\le C\Vert v{\Vert }_{\infty }/R$ on balls of arbitrarily large radius $R$. Letting $R\to \infty$ proves $Dv=0$, hence $u$ is constant. This proves the one-sided harmonic Liouville theorem in every dimension.