Cauchy sequence

In real analysis, a sequence $$\{a_n\}$$ is said to be Cauchy if the distance between any two elements in the sequence eventually get arbitrarily close. Formally, a sequence is Cauchy if, for every $$\varepsilon > 0$$, there exists some $$N$$ such that, for all $$n, m > N$$, $$d(a_n, a_m) < \varepsilon$$ where $$d(\cdot, \cdot)$$ is denoted as the metric.

Examples
The most natural example to look at is one that is familiar to many people. Consider the real numbers which is a metric space defined with the normal metric $$|\cdot |$$. A Cauchy sequence is a sequence in $$\mathbb R$$ with the property that, given any $$\varepsilon > 0$$, we can find some positive constant $$N$$ such that for any $$n, m > N$$, we have that $$|x_n - x_m| < \varepsilon$$.

As an example, take the sequence $$x_n = \frac{1}{n}$$. Fix some $$\varepsilon > 0$$. Choose $$N = \frac{2}{\varepsilon}$$. Then, for $$n, m > N$$, we have

$$|x_n - x_m| = \left|\frac{1}{n} - \frac{1}{m}\right| \leq \left|\frac{1}{n}\right| + \left|\frac{1}{m}\right| < \frac{1}{N} + \frac{1}{N} = \frac{2}{N} < \varepsilon.$$ Thus, $$\{x_n\}$$ is a Cauchy sequence.

Relations to complete metric spaces
A Cauchy sequence need not have a limit if the underlying space is not complete. However, if the space is complete, then every Cauchy sequence converges to a limit in the space. This, in a sense, is the definition of a complete metric space.

Important properties
Any Cauchy sequence in a complete metric space converges. This fact is extremely powerful and gives rise to many important theorems within real analysis.
 * Every convergent sequence is a Cauchy sequence (proof).
 * Every Cauchy sequence is bounded (proof).
 * Every Cauchy sequence in $$\mathbb R$$ converges – this comes immediately from the fact that $$\mathbb R$$ is complete with respect to the standard metric. In fact, one can generalize this to the following statement.