Does the extended real line form a metric space such that we can define limits in the usual way?
I read on a blog that if we define $$d(x,y) = \left|\,\overline{\arctan}(x) - \overline{\arctan}(y)\right|\;,$$
where
$$\overline{\arctan}(x)= \begin{cases} \pi /2 & \text{if } x=\infty \\ -\pi/2 & \text{if } x= -\infty \\ \arctan(x) & \text{else} \end{cases} $$
This clearly gives a metric space, but I'm having trouble showing that limits in this metric space correspond to the usual ones. Basically, I'm trying to explain why $\lim_{x \rightarrow \infty} f(x)$ has a definition which is different from $\lim_{x \rightarrow p} f(x)$.
