Cauchy Criterion for Uniform Convergence

Real Analysis · Axiom Academy

LESSON Cauchy Criterion for Uniform Convergence A test for uniform convergence that compares the functions to each other — so it works even when the limit is unknown. 1. Uniform Convergence, Measured by the Sup A sequence f_n converges uniformly to f on a set S when a single N works for every point at once: for all and all , . Packaged with the supremum, the whole condition collapses to one number shrinking to zero. N depends on only — never on x the sup distance is the worst gap over all of S 2. The Cauchy Criterion: Drop the Limit Here is the key move. Instead of comparing f_n to an unknown limit, compare two members of the sequence to each other . If every pair of late terms is uniformly close, the sequence must converge uniformly — and the converse holds too. 3. Uniform Cauchy vs Pointwise Cauchy The strength is entirely in the order of the quantifiers. Pointwise Cauchy fixes x first, then finds an N(x) — which may have to grow without bound as x moves. Uniform Cauchy demands one N before x is named, good for the whole set at once. For each fixed x : given , some N(x) makes . The index N may depend on x . One N serves all x simultaneously: . The index is independent of x . 4. Worked Example: Prove It Without the Limit Take this sequence of partial sums on [0,1] . We will prove it converges uniformly using only the Cauchy criterion — at no point do we need to know what it converges to. Fix and any . The difference is exactly the chunk of terms between them:

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