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")

EDIT: In his 1994 paper "$\sigma$-centred forcing and reflection of (sub)metrizability" in TAMS (MR1179593 94g:54003), Frank Tall writes:

"It is well known (proved by the same method that proves the product of $\le 2^{\aleph_0}$ separable spaces is separable) that the finite support iteration of $\le 2^{\aleph_0}$ $\sigma$-centred orders is $\sigma$-centred."