Cauchy Sequence

Definition

For a measurement space \((X, d)\) where \(X\) is the set of points and \(d\) is a distance measure (distance function) of the space, sequence \(S=(x1, x2, x3, ....., xn, ...),\) is a Cauchy sequence if

for any positive integer \(\epsilon\), there exists a bound \(N\) such that \(d(xa, xb) < \epsilon\) for any indices \(a, b > N\).

Intuitive Explanation

First, it is obvious that sequence \(S\) converges. Unformally speaking, we call a sequence to be a converge sequence if the number in the sequence stablizes to a fixed number, e.g. 0.9, 0.99, 0.999, 0.9999, … seems to converge to 1.

For more general situations, the distance function may not be the absolute distance. For example, we may also want to generalize the concept of convergence to 2-d points. Therefore, the definition of convergence here requires the definition of some distance function \(d\).

Similar to the definition of the extreme point (or limit), the definition of convergence also uses the \(\epsilon-\delta\) language. We can unformally translate the \(\epsilon-\delta\) languae in the Cauchy sequence definition to the following:

\(\lim xn = x*\) as \(n \rightarrow \infty\).