Completeness of The Real Numbers - Forms of Completeness - Cauchy Completeness

Cauchy completeness is the statement that every Cauchy sequence of real numbers converges.

The rational number line Q is not Cauchy complete. An example is the following sequence of rational numbers:

Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.)

Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers.

In mathematical analysis, Cauchy completeness can be generalized to a notion of completeness for any metric space. See complete metric space.

For an ordered field, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the Archimedean property taken together are equivalent to the others.

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