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. 

(EDIT: In the first version of the question I forgot to mention that it was  Stefan Geschke who suggested that there should be a proof similar to "the product of continuum many separable spaces is still separable")