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For each nonzero integer tuple $k=(k_1,\dots ,k_{n+1})$, putIf its first $n$ coordinates are all zero, then $k_{n+1}\ne 0$ and $H_k$ is empty. Otherwise it is a closed proper affine hyperplane. It has empty interior: from any of its points, an arbitrarily small displacement in the nonzero normal direction $(k_1,\dots ,k_n)$ leaves it. Thus every $H_k$ is nowhere dense.

There are countably many integer tuples, and ${\mathbb{R}}^n$ is complete. The Baire category theorem therefore makes the complement of their union dense and nonempty. Choosing any point there givesEquivalently, $1,x_1,\dots ,x_n$ are linearly independent over $\mathbb{Q}$: multiplying a rational relation by a common denominator produces just such an integer relation. This is an instance of Baire avoidance of countably many affine hyperplanes.