Egoprox Sequences
Why
We want to talk about sequences in a metric space which are “bunching up,” and whether and when this means they converge. We generalize the notion of real egoprox sequences.
Definition
A sequence in a metric space is egoprox (or Cauchy) if for every positive real number, there exists a final part of the sequence so that any two terms are less than the positive number apart.1
Notation
Suppose $(X, d)$ is a metric space and
$\seq{x}$ is a sequence in $X$.
$\seq{x}$ is egoprox means for every $\epsilon
> 0$, there exists $N \in \N $ so that,
for all $m, n \geq N$
\[
d(x_n, x_m) < \epsilon
\]
- The term Cauchy is universal, but in accordance with the Bourbaki project’s guidelines on naming, we will tend to use the term egoprox. ↩︎