Let u be the solution of an ordinary boundary value problem and $P^\pi u$ the Galerkin projection on a space of piecewise polynomial functions of degree $ \leqq r$. We are going to prove that the following estimate holds with $L_\infty $-norms; \[ \left\| {P^\pi u - u} \right\|_{[0,1]} \leqq C \mathop {\max }\limits_i h_i^{ * r + 1} \left\| {u^{(r + 1)} } \right\|I_i^ * .\] The mesh need not be quasiuniform, but it is not completely unconstrained in all cases. $I_i^ * $ is the union of a subinterval $I_i $ and some intervals in its neighborhood the number of which is independent of the mesh. The length of $I_i^ * $ is denoted by $h_i^ * $.