What Cauchy sequences can we find in
with (british-)rail metric?
First case would be sequences on any line
Are there any other? Balls with center at
are the same as in
with euclidean metric. So maybe that would be all sequences approaching
The British Rail metric (or the SNCF metric) is not
the taxicab metric.
The taxicab metric, being a metric induced by a norm on
(namely the 1-norm), is Lipschitz equivalent to the usual metric, so their Cauchy sequences agree.
On the other hand, the only Cauchy sequences in the SNCF metric is, as you have found, those Cauchy sequence in the usual metric that converges to 0, or eventually lie on a line.