Cauchy Criterion (Sequence)

analysis

Definition

Cauchy Criterion (Sequence)

A sequence $(a_n{)}_{n\ge 0}$ of real numbers is convergent if and only if it is a Cauchy sequence.

Formally, the sequence converges if for every $\epsilon >0$, there exists an index $N(\epsilon )$ such that:

$$\forall n,m>N(\epsilon ):|a_n-a_m|<\epsilon$$

Visual Intuition

In a Cauchy sequence, the terms eventually become arbitrarily close to each other.

Significance

The Cauchy criterion is powerful because it allows one to prove the convergence of a sequence without knowing the value of the limit beforehand. It is equivalent to the completeness of the real numbers.