Metric Spaces, Cauchy Sequence

Cauchy Sequence

A sequence s in a metric space is Cauchy (biography) if every ε implies an n such that all points beyond s_n are within ε of s_n. The tail of the sequence can always be contained in a small open set about s_n.

You may have seen another definition of Cauchy: any two points at or beyond s_n are within ε of each other. In particular, s_n and s_j are within ε of each other, hence this definition implies the previous. Conversely, if p and q are within ε/2 of s_n, they are within ε of each other by the triangular inequality. The two definitions are equivalent; use which ever one you like.

Real functions, defined for x ≥ 0, can also be Cauchy. Every ε implies y such that |f(x),f(y)| < ε for x ≥ y. Most of the time we deal with sequences.

Set ε to 1 to show that a Cauchy sequence, or the tail of a Cauchy function, is bounded.

If there is a smallest accessible positive distance ε in our metric space, restrict a Cauchy sequence to ε/2 to show that it becomes constant. Clearly such a sequence converges to s_n.

In fact, every convergent sequence is Cauchy. If p is the limit point, find n such that everything beyond s_n is within ε/2 of p. By the triangular inequality, everything beyond s_n is within ε of s_n.

A metric space is complete if every Cauchy sequence converges to some point in the space. In other words, a sequence is Cauchy iff it is convergent.

Before we apply this to metric spaces in general, let's show that the reals are complete. After all, metric spaces are based on distance, which is measured using real numbers, so we'd better get a handle on the reals first.