Perturbing All Points in R^d into a General Position

We prove that there is a way to move each point in \mathbf{R}^d less than a prescribed amount, so that no two points are moved to the same point, and the resulting set of points are in a general position (i.e. no three points are in a line, no four points are in a plane, and so on). Our proof is based on transfinite recursion and induction.

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