Show that a metric space is not complete under a given metric?

Question

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?

Answer 1

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

Answer 2

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 $\mathbb{R}$.