= Convergence from arbitrary variable-lag increments
{title2=$a_{n+h(n)}-a_n\to0\text{ for every }h:\mathbb N_{>0}\to\mathbb N_{>0}$}
A real <sequence> converges if and only if the displayed condition holds. A <Cauchy sequence> makes all future differences uniformly small, so the forward implication is immediate. Conversely, failure of the <Cauchy sequence> criterion produces a fixed tolerance and, for each $n$, a future term differing from $a_n$ by at least half that tolerance. Choosing the least such positive lag defines one function that contradicts the displayed condition. The <completeness of the real numbers> then gives convergence. Testing only every fixed integer lag is weaker: $a_n=\log n$ passes those tests but diverges.
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