In this paper we obtain a modified version of the classical theorem of Pontrjagin and Schnirelmann concerning the covering dimension of a compact metric space and the box-counting dimensions associated with the metrics on the space. T. Miyata and T. Watanabe defined box-counting dimension for approximate sequences X which represent compact metric spaces X as approximate resolutions p : X → X. We show that the covering dimension equals the infimum of the box-counting dimensions associated with the approximate resolutions of the space.