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I have to show that the metric space $M = (\mathbb{R},d)$ is not complete. The given metric is $d = |\tanh(x)-\tanh(y)|$.

I know the definition of completness using Cauchy sequences and I assume the incompleteness has something to do with the fact that $\tanh(x)$ has two limits. So how exactly can I show that $M$ is not complete?

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The sequence $a_n=n$ is a Cauchy sequence that does not converge in $M$.

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We have the identity:

$|\tanh(x) - \tanh(y)| = |\tanh(x-y)| \times |1-\tanh(x)\tanh(y)|$

Consider the sequence $\{a_n\}$ given by $a_i=i$.

Note that $|\tanh(x)| < 1$ and $\lim_{x \to +\infty} \tanh(x) = 1$

We have:

$|\tanh(m+k) - \tanh(m)| = |\tanh(k)| \times |1-\tanh(m)\tanh(k)| \leq |1-\tanh(m)\tanh(k)|$

This could be made as small as desired when $m,k \to \infty$. Therefore, the sequence given above is Cauchy. But it approaches infinity that is not in $\mathbb{R}$.

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  • $\begingroup$ nice, thanks a lot! $\endgroup$ – Philipp Nov 11 '17 at 13:51
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    $\begingroup$ @Philipp: You're welcome. Notice that if you change $\tanh$ to $\tan^{-1}$ and use the distance $\mathrm{d(x,y)} = |\tan^{-1}(x)-\tan^{-1}(y)|$ instead, $\mathbb{R}$ will still be not complete with this new metric. Try to prove it. $\endgroup$ – stressed out Nov 11 '17 at 14:00

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