= Uniqueness of best uniform polynomial approximation
For a fixed degree bound, two best approximating <polynomials> have a best approximating average. At each active point of that average their errors agree, hence the two polynomials agree. There must be at least two more active points than the degree: otherwise <polynomial interpolation> constructs a strictly improving sign-matching perturbation. The difference has too many zeros and must vanish. Finite-dimensional coefficient compactness also gives existence.
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