The Cauchy Criterion

Real Analysis · Axiom Academy

A test for convergence that never names the limit: the terms get arbitrarily close to each other . 1. Close to Each Other, Not to a Limit A sequence is Cauchy when, no matter how small a tolerance you pick, there is a cutoff index N beyond which any two terms differ by less than . The phrase "for all " is the whole point — we measure the distance between a pair of tail terms, not their distance to some limit we'd have to know in advance. 2. One That Passes, One That Fails The criterion is a clean pass/fail filter. Compare the harmonic terms with the alternating a_n=(-1)^n : one bunches up, the other never does. The animation runs both at once so the difference is visible in the motion itself. Terms crowd together. Past index N the gap is at most , so for any choose and . Terms stay split. Consecutive terms always satisfy |a_ n+1 -a_n|=2 , so no N can push the gap below, say, . 3. In , Cauchy Means Convergent Here is the payoff. The shrinking tail-bands are nested boxes closing in on a single location. In that location is always a real number, so the sequence converges — the criterion certifies a limit it never had to name. This property is called completeness . Every convergent sequence is Cauchy: if , two terms near L are near each other. This holds in any setting. In the implication reverses: every Cauchy sequence has a real limit. The nested tails trap a point, and in that point exists.

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