Why write $[a,+\infty)$ instead of $[a,\infty)$?

Question

I have seen $[a,+\infty)$ written and also $[a,\infty)$, for some $a \in \mathbb{R}$.

Why would someone want to write $[a,+\infty)$ instead of just $[a,\infty)$?

Answer 1

Usually whenever $+\infty$ is written in lieu of merely $\infty$, it seems mostly meant to just clarify that "this is the positive infinity". Typically when left signless, people tend to assume $\infty$ refers to the positive one anyways though. It can also be used to distinguish from the element added to the extended real/complex numbers if the situation necessitates it.

Granted for instances like yours it may often be easily understood - for example, at least in my experience, $[a,b)$ implicitly has $a<b$, so if $a \in \Bbb R$, then it shouldn't be ambiguous what $\infty$ refers to in $[a,\infty)$.

But eh, it's one keystroke for a little extra clarity, and some authors simply have their own quirks, so it doesn't hurt a ton either, you know?

Answer 2

There are compactifications of $\mathbb{R}$ that only use one infinity point ($\infty$) (the numberline would "look like" an infinite radius circle) with the positives and negatives connected at both $0$ and $\infty$

So I think the main reason why the "extended numberline" $[-\infty, +\infty]$ uses the $+\infty$ symbol is to state the difference clearly. However, as the concept of inifinity is used so often in limits, I'd say that it's just fine to use $\infty$ meaning the same as $+\infty$.

Also note that, when working with sequences on the natural numbers, $\infty$ is almost universally used