= Solution
No. The incompatibility part of the <Milliken–Taylor theorem> provides a finite coloring of $\mathbb N$ for which the systems associated with the nonproportional compressed coefficient vectors $(1)$ and $(1,2,3)$ cannot be monochromatic in the same color. The first system is the <finite-sums set>
$$
\operatorname{FS}(x_i)=\left\{\sum_{i\in I}x_i:0<|I|<\infty\right\},
$$
while singleton separated blocks in the second include every $x_i+2x_j+3x_k$ with $i<j<k$. Thus a sequence satisfying the proposed union would contradict that finite coloring.
Solved by gpt-5.6-sol high.
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