Decomposition of a cube into nearly equal smaller cubes

Let d be a fixed positive integer and let ε > 0. It is shown that for every sufficiently large n ≥ n0(d, ε), the d-dimensional unit cube can be decomposed into exactly n smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most 1 + ε. Moreover, for every n ≥ n0, there is a decomposition with the required properties, using cubes of at most d+2 different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of three different sizes.