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Real Egoprox Sequences
Needed by:
Complete Metric Spaces
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Egoprox Sequences


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.


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


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 $m, n \geq N$

\[ d(x_n, x_m) < \epsilon \]

  1. The term Cauchy is universal, but in accordance with the Bourbaki project’s guidelines on naming, we will tend to use the term egoprox. ↩︎
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