About dense subsets of Tychonoff products of discrete spaces

The Hewitt-Marczewski-Pondiczery theorem states that if X= is the Tychonoff product, where d(Xα)≤τ≥ω for all α∈A and |A|≤2τ, then d(X)≤τ.We prove that in the product of discrete spaces Dα=ω there is a dense countable subset Q=∪{Qk:k∈ω}, the union of disjoint finite sets Qk, satisfying the following conditions:-if F⊆Q is such that |F∩Qk|≤1 for all k∈ω, then F is a discrete closed in set;-if C⊆ω is infinite and for F⊆Q there is m0∈ω such that |Qk∩F|≤m0 for all k∈ω, then ∪{Qk:k∈C}∖F is dense in ;-Q contains no convergent in non-trivial sequences;