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Use of the Axiom of Choice is unnecessary here. One can write down an explicit $\mathbb{Q}$-linearly independent subset $\mathcal{T}$ of $\mathbb{R}$ with size $2^{\aleph_0}$ as I wrote in this answer. If it so happens that some of the $c(n)$ are in the $\mathbb{Q}$-span of this set $\mathcal{T}$, then remove the finitely or countably many elements which are involved in expressing the $c(n)$ as a $\mathbb{Q}$-linear combination of elements of the original set $\mathcal{T}$; what remains of $\mathcal{T}$ still has size $2^{\aleph_0}$. The $\mathbb{Q}$-span of this remainder has the desired property. The set in question will be a Borel set (of Lebesgue measure 0).