We derive an a priori error estimate and thereby prove convergence for the multiscale method first presented by Larson and Malqvist in ”Adaptive variational multiscale methods based on a posteriori error estimation” [CMAME,196,2007,2313-2324].The proof strongly relies on the local behavior of the elliptic differential operator on fine scales.We track the decay rate of the fine scale basis functions for arbitrary positive bounded diffusion coefficient.The decay rate is the key result which leads to an a priori bound of the error in the multiscale approximation.We present numerical examples that confirms the theoretical results.