For questions relating to the properties of Cauchy sequences.

A sequence $\{x_n\}$ in an arbitrary metric space, and in particular the space $\Bbb{R}$, is called Cauchy if the terms of the sequence become arbitrarily close together; that is, for every $\epsilon > 0$, there exists an $N$ such that

$$n, m \ge N \implies d(x_n, x_m) < \epsilon$$

where $d$ is the distance function for the metric space. In the particular case of the real numbers, this condition becomes

$$n, m \ge N \implies |x_n - x_m| < \epsilon$$

A complete metric space is a metric space in which every Cauchy sequence is convergent; this gives an alternate definition of convergence of a sequence that does not rely on the limiting value.

Source: the Cauchy sequence article on Wikipedia.

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