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 Post subject: Cauchy sequence, British Rail metric.Posted: Sat, 21 May 2011 11:23:12 UTC
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Joined: Sun, 24 Apr 2011 18:50:23 UTC
Posts: 106
Location: Poland
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 point?

Last edited by KamilJ on Sat, 21 May 2011 17:17:43 UTC, edited 1 time in total.

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 Post subject: Re: Cauchy sequence, taxi-cab metric.Posted: Sat, 21 May 2011 15:04:14 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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KamilJ wrote:
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 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 (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.

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 Post subject: Re: Cauchy sequence, taxi-cab metric.Posted: Sat, 21 May 2011 17:21:22 UTC
 Senior Member

Joined: Sun, 24 Apr 2011 18:50:23 UTC
Posts: 106
Location: Poland
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.

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