Cauchy sequence

A sequence is Cauchy when its terms become arbitrarily close to one another beyond some index.

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.

A Cauchy subsequence is a subsequence that is a Cauchy sequence. Every convergent subsequence is Cauchy.

Every Cauchy sequence of real numbers converges to a real number.

The least-upper-bound property states that every nonempty set of real numbers that has an upper bound has a supremum in the real numbers. It is an order-theoretic formulation of the completeness of the real numbers.