A nonexpansive bijection from a ball between c₀₀ and c₀ onto a ball between c₀₀ and ℓ∞ moves each coordinate inward along a signed permutation

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A nonexpansive bijection from a ball between c₀₀ and c₀ onto a ball between c₀₀ and ℓ∞ moves each coordinate inward along a signed permutation · Prove2Me