I am looking for a proof (or better, a reference) of the following fact: 

The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than $(2^{\aleph_0})^+$ steps. 

(Assuming it is true.  Isn't there a proof using the fact that the product of continuum many separable spaces is still separable? Or at least using the same idea.)