= Solution
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> give
$$
u(x)=\frac1{\omega_n\rho^n}\int_{B_\rho(x)}u
\leq\frac1{\omega_n\rho^n}\int_{B_{3\rho}(z)}u=3^nu(z).
$$
Taking the supremum over $x$ and the infimum over $z$ proves the <Harnack inequality for harmonic functions>:
$$
\boxed{\sup_{B_\rho(y)}u\leq3^n\inf_{B_\rho(y)}u.}
$$
This includes a zero infimum, which forces $u=0$ on the smaller ball.
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