Do projections stay close together?

We estimate the rate of convergence of products of projections on K intersecting lines in Rd. More generally, consider the orbit of a point under any sequence of orthogonal projections on K arbitrary lines in Rd. Assume that the sum of the squares of the distances of the consecutive iterates is less than ε. We show that if ε tends to zero, then the diameter of the orbit tends to zero uniformly for all families L of a fixed number K of lines. We relate this result to questions concerning convergence of products of projections on finite families of closed subspaces of ℓ2.