Let U\subset \mathbb{R}^n be bounded and connected, and u\in C^2(U)\cap C(\overline U) with \Delta u=0 in U. Then \max_{\overline U}u = \max_{\partial U}u and \min_{\overline U}u = \min_{\partial U}u. In particular, if u attains an interior maximum or minimum, then u is constant.