= Diagonal compactness for bounded Lipschitz functions
On a separable <metric space>, a sequence uniformly bounded in <bounded Lipschitz norm> has a pointwise convergent subsequence. Enumerate a countable dense subset and use the <Bolzano-Weierstrass theorem> and <diagonal subsequence argument> to obtain convergence there. Uniform control of the <Lipschitz constant> then makes the subsequence Cauchy at every point. The <pointwise closure of the bounded Lipschitz unit ball> preserves the original norm bound after rescaling.
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