In a similar way one can define Cauchy sequences of rational or complex numbers.

There is also a concept of Cauchy sequence in a group :

A metric space is called complete if all Cauchy sequences converge.

Completeness can be proved in a similar way to the construction from the Cauchy sequences.

The last inequality proves that (y) is a Cauchy sequence.

A metric space is complete if every Cauchy sequence in it converges.

"Because Cauchy sequences require the notion of distance, they can only be defined in a metric space."

The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.

This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0.

Therefore, we can show is a Cauchy sequence and thus it converges to a point .

One completes the rationals by adding the limit of all Cauchy sequences to the set.

Another approach is to define a real number as the "'limit of a Cauchy sequence of rational numbers"'.

A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric.

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large.

Let R be the set of Cauchy sequences of rational numbers.

Two Cauchy sequences are called "equivalent" if and only if the difference between them tends to zero.

More precisely, it is reflexive since the sequences are Cauchy sequences.

In metric spaces, one can define Cauchy sequences.

Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number.

Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.

It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges.

Notice that every convergent sequence is a Cauchy sequence.

There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences.

By definition, in a Hilbert space any Cauchy sequence converges to a limit.

In analysis, Cauchy sequence.