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The question is in the title:

Is there a (extended) metric on the extended reals which yields regular and infinite limits?

but in particular I want know the explicit construction of said metric.

Of course, by "yields regular and infinite limits" I mean: Let $\overline{\mathbb{R}} = \mathbb{R} \cup \{\pm \infty\}$. The (extended) metric on $\overline{\mathbb{R}}$ is so that: Given a sequence $a : \mathbb{N} \to \overline{\mathbb{R}}$ , with only real numbers in its image, we have: $$\lim a = a_\infty \in \mathbb{R}\Leftrightarrow \forall \epsilon > 0 : \exists N\in \mathbb{N} : \forall n \in \mathbb{N}, n\geq N : |a_\infty - a_n| < \epsilon$$ $$\lim a = \infty \Leftrightarrow \forall c > 0 : \exists N \in \mathbb{N} : \forall n \in \mathbb{N}, n\geq N : a_n > c$$ $$\lim a = -\infty \Leftrightarrow \forall c > 0 : \exists N \in \mathbb{N} : \forall n \in \mathbb{N}, n\geq N : a_n < c$$

Here $\lim a$ refers to he limit in the aforementioned metric space.

This is hinted here, but I still do not know how to find the metric in question.

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  • $\begingroup$ It doesn't extend the usual metric, but you can take a homeomorphism with $[-1,1]$ and use the resulting metric. This preserves the topology and hence limits. $\endgroup$ – Matt Samuel Jan 23 '16 at 21:47
  • $\begingroup$ @Matt Samuel How do I get the explicit metric from this? $\endgroup$ – Stefan Perko Jan 23 '16 at 21:51
  • $\begingroup$ Use an explicit homeomorphism, say $x\mapsto x/(1+x)$, then $d(x,y)=|x/(1+x)-y/(1+y)|$. $\endgroup$ – Matt Samuel Jan 23 '16 at 21:53
  • $\begingroup$ Actually $x/(1+|x|)$ would be a better choice because the other one isn't a homeomorphism. $\endgroup$ – Matt Samuel Jan 23 '16 at 21:55
  • $\begingroup$ @MattSamuel This may be a dumb question: How do you evaluate the metric at say $(\infty, x)$ for some $x$? (How do you deal with $\pm \infty$?) $\endgroup$ – Stefan Perko Jan 23 '16 at 22:00
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As hinted in your link, there is a bijection $f:\overline{\Bbb R} \to [-\frac \pi 2,\frac \pi 2]$ given by $$f(x)=\begin {cases} -\frac \pi 2 & x=-\infty\\\arctan x & -\infty \lt x \lt \infty \\\frac \pi 2 & x=+\infty \end {cases}$$ Now your metric on $\overline{\Bbb R}$ is $d(x,y)=|f(x)-f(y)|$

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