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or counterexample
Goldstern
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Finite support iterations of $\sigma$-centered forcing notions

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.)

(A counterexample would be welcome, too.)

Goldstern
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