Definition:Cauchy Sequence

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Definition

Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$.


Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: d \left({x_n, x_m}\right) < \epsilon$


Normed Vector Space

Let $\left({V, \left\Vert{\,\cdot\,}\right\Vert}\right)$ be a normed vector space over a normed division ring $\left({R, \left\Vert{\,\cdot\,}\right\Vert_R}\right)$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $V$.

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left \Vert{x_n - x_m}\right \Vert < \epsilon$


Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring

Let $\sequence {x_n}$ be a sequence in $R$.

Then $\sequence {x_n} $ is a Cauchy sequence in the norm $\norm {\, \cdot \,}$ if and only if:

$\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$

that is,

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$


Standard Number Fields

Complex Numbers

Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.


Then $\left \langle {z_n} \right \rangle$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{z_n - z_m}\right| < \epsilon$

where $\left|{z_n - z_m}\right|$ denotes the complex modulus of $z_n - z_m$.


Real Numbers

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.


Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$


Rational Numbers

The concept can also be defined for the set of rational numbers $\Q$:


Let $\left \langle {x_n} \right \rangle$ be a rational sequence.


Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$

where $\Q_{>0}$ denotes the set of all strictly positive rational numbers.


Cauchy Criterion

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.


Also see

Thus in $\R$ and $\C$ a Cauchy sequence and a convergent sequence are equivalent concepts.

  • Results about Cauchy Sequences can be found here.


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources