Cauchy sequence, British Rail metric.

KamilJ · **Posted:** Sat, 21 May 2011 11:23:12 UTC

What Cauchy sequences can we find in $(\mathbb{R}^2 , d_r)$ with (british-)rail metric?

First case would be sequences on any line y=ax

.
Are there any other? Balls with center at (0,0)

are the same as in $\mathbb{R}^2$ with euclidean metric. So maybe that would be all sequences approaching (0,0)

point?

outermeasure · **Posted:** Sat, 21 May 2011 15:04:14 UTC

KamilJ wrote:

What Cauchy sequences can we find in $(\mathbb{R}^2 , d_r)$ with (british-)rail metric?

First case would be sequences on any line y=ax

.
Are there any other? Balls with center at (0,0)

are the same as in $\mathbb{R}^2$ with euclidean metric. So maybe that would be all sequences approaching (0,0)

point?

The British Rail metric (or the SNCF metric) is not the taxicab metric.

The taxicab metric, being a metric induced by a norm on $\mathbb{R}^2$ (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.

KamilJ · **Posted:** Sat, 21 May 2011 17:21:22 UTC

outermeasure wrote:

The British Rail metric (or the SNCF metric) is not the taxicab metric.

I'm sorry, I typed the wrong title by mistake.

outermeasure wrote:

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.

Thank you for confirmation.

S.O.S. Mathematics CyberBoard

Cauchy sequence, British Rail metric.

Who is online