Proportional Dvoretzky-Rogers factorization result

If X is an n-dimensional normed space and epsilon is an element of (0, 1), there exists m greater than or equal to (1 - epsilon)n, such that the formal identity i(2,infinity):l(2)(m) --> l(infinity)(m) can be written as i(2,infinity) = alpha o circle beta,beta: l(2)(m) --> X, alpha: X --> l(infinity)(m), with \\alpha\\ . \\beta\\ less than or equal to c/epsilon. This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.