Can we study sequences in the extended real numbers?

Question

Is there any significance to study sequences in $\Bbb R \cup \{-\infty,\infty\}$ that is functions $f : \Bbb N \to \Bbb R \cup \{-\infty,\infty\}$. I mean do we define and study convergence of such sequences? I mean such sequences like $1,\infty,3,1.5,5,2.3,\pi,........$ studied?

Answer 1

The map $(-1,1)\ni x\mapsto \frac{x}{1-\vert x\vert}:=f(x)\in\mathbb{R}$ may bei extended to a bijection $F\colon [-1,1]\to \Bbb R \cup \{-\infty,\infty\}$ by $F(x):=f(x)$ for $-1<x<1$ and $F(\pm1):=\pm\infty$. Then, when defining a distance on $\Bbb R \cup \{-\infty,\infty\}$ by $d(u,v):=\vert F^{-1}(u)-F^{-1}(v)\vert$, you make $\Bbb R \cup \{-\infty,\infty\}$ a metric space. And your sequences how are nothing else but sequences in a (special) metric space.

Answer 2

Sure. Take the topology as the order topology, and say that a sequence $\{ x_n \}$ converges to $L$ if, given any open set $O$ containing $L$, there is a suitably large $N$ such that

$$x_n \in O$$

for all

$$n > N$$

.