Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space

Question

I have to show that the real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space.

Certainly, I have to search for a cauchy sequence of real numbers with respect to given metric that must not be convergent. But I am unable to figure out that. Can anybody help me with this.

Thanks for helping me.

Answer 1

Consider a sequence that grows without bound. Such a sequence isn't Cauchy in the usual metric on $\mathbb{R}$, but will be under this metric.

Answer 2

Consider $x_n = n$. Let $\varepsilon > 0$ and choose $\displaystyle N > \tan\bigg(\frac\pi2 - \varepsilon\bigg)$. If $m, n > N$ then $\displaystyle \{\arctan m, \arctan m\} \subseteq \bigg(\arctan N, \frac\pi2\bigg)$. Thus $$d(x_m, x_n) = \vert \arctan m - \arctan n \vert \leq \bigg \vert \frac\pi2 - \arctan N \bigg\vert < \bigg \vert \frac\pi2 - \frac\pi2 + \varepsilon \bigg\vert= \varepsilon$$ Thus $(x_n)$ is a Cauchy sequence. Observe that as $n \to \infty$, $x_n \to \pi/2$. But $(x_n)$ does not converge to any element in $\mathbb R$ since there is no $x \in \mathbb R$ such that $\arctan x = \pi/2$.