Learn Turn common continuity control into uniformly convergent subsequences, distinguish a family from its closure, and justify optimization over functions.
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The extraction proof must obtain one error cutoff for every input.
One index cutoff must control the error at all inputs simultaneously. Separate cutoffs at each input need not have a finite common maximum.
Finitely many centers will turn separate convergence bounds into one uniform bound.
A finite cover by balls at that scale, with centers in the compact metric space. The finite number of centers is essential.
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A sharp maximum, an operator image and a grid certificate
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Distinguish local family-wide continuity from uniform control, and turn compactness and pointwise bounds into rigorous finite-grid certificates.
6 practice tasks.
Build one subsequence that converges at a countable dense set, then use equicontinuity and finite nets to control every input simultaneously.
6 practice tasks.
Prove the converse compactness conditions, pass to a compact closure, and justify optimization over closed function constraints.
6 practice tasks.