Because by the maximum principle the function $\displaystyle u$ assumes it maximum and minimum on the boundary, $\displaystyle S$. Since it constatly assumes zero it must mean the harmonic function is identitically zero on $\displaystyle V$. As a corollary, we can show that the Dirichlet problem has at most one solution. Because if $\displaystyle \xi$ and $\displaystyle \mu$ are harmonic on an open set and solve the same Dirichlet problem then $\displaystyle \xi - \mu$ is also harmonic, and furthermore $\displaystyle \xi - \mu$ is identitcally zero on the boundary. Therefore, $\displaystyle \xi = \mu$ on all points in open set.