Are there any metric spaces with no Cauchy sequences?

[dumb_curiosity]

A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default?

When trying to think of a space with no cauchy sequences, I invision a space where points are sufficiently far apart from each other. Since all metric spaces are T1,T2,T3, and T4, I get the idea that you can "separate" points, and closed sets in general in a metric space. That's what made me think about a metric space where points are so far apart, that there are no cauchy sequences.

 

[WWGD]

You can always define Cauchy sequences by repeating a point : {p,p,...,p,...} . In discrete (and other) spaces , only eventually-constant sequences converge. So the best is to say that sequences without repeated terms may not converge.