= Baire avoidance of countably many affine hyperplanes
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Every proper <affine hyperplane> in $\mathbb R^n$ is closed and has empty interior: a small displacement along its nonzero normal vector leaves it. The <Baire category theorem> therefore makes the complement of a countable collection of such hyperplanes dense. In particular, avoiding the hyperplanes $\sum_{j=1}^n k_jx_j=k_{n+1}$ for nonzero integer tuples yields a vector for which $1,x_1,\ldots,x_n$ are linearly independent over $\mathbb Q$. Tuples with all the first $n$ entries zero define an empty forbidden set.
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