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In a metric space, is it possible to find a sequence which converges to two different limits with respect to two different metrics?

Obviously the metrics can't be equivalent.

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  • $\begingroup$ What if we take one ae usual metric and the other one on the lines d(x,y) =1 for $x \ne y$ in the second one, the sequence will eventually be constant $\endgroup$
    – Shailesh
    Oct 1, 2015 at 12:49
  • $\begingroup$ @Shailesh, but they won't converge to two different limits, then, right? Discrete metric won't ever work. $\endgroup$ Oct 1, 2015 at 13:16
  • $\begingroup$ Yes, I realised that after posting the comment. Thanks $\endgroup$
    – Shailesh
    Oct 1, 2015 at 13:34

1 Answer 1

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With two different metrics? Yes, obviously. (But with two different metrics it is not the same metric space, by definition -- the concept of a metric space includes which metric we're using).

For example take $\mathbb R$ with respectively the standard metric, and the metric $$ d_2(a,b)=|f(a)-f(b)| \quad\text{where }f(x)=\begin{cases} \pi & \text{if }x=0 \\ 0 & \text{if }x=\pi \\ x & \text{otherwise} \end{cases} $$

Then $a_n=\frac1n$ converges to $0$ in the usual metric but to $\pi$ in the $d_2$ metric.

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    $\begingroup$ That's a great example and exactly the kind of thing I wanted. Thanks. $\endgroup$ Oct 1, 2015 at 13:19
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    $\begingroup$ Is there a name for the type of metric that $d_2$ is? (At first I thought this could not be a metric, but it surprised me to see that it really is...) $\endgroup$
    – user193810
    Oct 1, 2015 at 14:13
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    $\begingroup$ @Pakk It is just the usual metric on $\mathbb{R}$, with 0 and $\pi$ swapped. $\endgroup$
    – Yakk
    Oct 1, 2015 at 15:44
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    $\begingroup$ @Pakk: Whenever $(X,d_X)$ is a metric space and $f:Y\to X$ is an injection, you can make $Y$ into a metric space by defining $d_Y(a,b)=d_X(f(a),f(b))$. In the particular case where $Y\subseteq X$ and $f$ is the inclusion map, this gives the subspace metric, but I don't recall seeing any nice name for the general case. One might say something like "the metric on $Y$ induced by $f$", though. $\endgroup$ Oct 1, 2015 at 18:22

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