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It is fairly easy to prove that $(\mathbb{R},d)$ is a complete space with d(x,y)=|x-y|:

  1. Take a Cauchy sequence and prove it is bounded.
  2. By the Bolzano-Weierstrass, every bounded sequence in $\mathbb{R}$ has a convergent subsequence. Let $\ell$ be this limit.
  3. Show that the Cauchy sequence converges towards $\ell$, using the triangular inequality, the fact that it is Cauchy, and the fact that it has a convergent subsequence.

I also know $(\mathbb{R},d)$ is not a complete space with $d(x,y)=|\arctan(x)-\arctan(y)|$.

What I do not understand is why the previous proof does not work here. In other words, where does the explicit expression of $d(x,y)$ come into account in the proof?

Kuifje
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1 Answers1

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The second step fails: the Bolzano-Weierstrass theorem doesn’t hold for the arctangent metric $d$. For instance, the sequence of positive integers is bounded in the metric $d$ but has no convergent subsequence. It’s important to realize here that the notion of boundedness that’s required is boundedness in the metric being used.

Brian M. Scott
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  • But the Cauchy sequence remains bounded right? Otherwise step 1 would already fail. – Kuifje Sep 23 '15 at 14:53
  • @Kuifje: Yes, that’s correct. In any metric space a Cauchy sequence is bounded with respect to the metric. – Brian M. Scott Sep 23 '15 at 14:57
  • @ Brian M. Scott: I understand that the sequence of positive integers has no convergent subsequence with the $d$ metric, but in a more general perspective, how can we determine if the Bolzano-Weierstrass theorem holds or not (given a metric)? If the Cauchy sequence is bounded in the metric being used, why would it fail? – Kuifje Sep 23 '15 at 15:11
  • @Kuifje: In general you should assume that the B-W theorem does not hold. I’m not sure what it refers to in your last question. – Brian M. Scott Sep 23 '15 at 15:16
  • @ Brian M. Scott: Thanks. (Just to be clear $it$ referred to B-W). – Kuifje Sep 23 '15 at 15:30
  • @Kuifje: You’re welcome. (I thought that it probably did, but I wanted to be sure.) – Brian M. Scott Sep 23 '15 at 15:30