Definition:Cauchy Criterion

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Definition

The Cauchy criterion is the necessary and sufficient condition for an infinite sequence in a complete metric space to converge that the absolute difference between successive terms with sufficiently large indices tends to zero.


It can be applied in a number of contexts:


Sequences

The Cauchy criterion is the condition:

For any (strictly) positive real number $\epsilon \in R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$.


Informally:

For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked.


Or to put it another way, the terms get arbitrarily close together the farther out you go.


Products

Let $\struct {\mathbb K, \norm {\,\cdot\,}}$ be a valued field.

Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ satisfies Cauchy's criterion if and only if:

$\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m\geq n \ge N: \norm {\prod_{k \mathop = n}^m a_n - 1} < \epsilon$


Also known as

Some sources refer to the Cauchy criterion as the Cauchy condition, but the same term is used for the Cauchy boundary condition.

Hence in order to reduce possible confusion, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the use of Cauchy criterion


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources