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

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 $$[a,\infty)$$?

## marked as duplicate by Yanior Weg, YiFan, José Carlos Santos, Lee David Chung Lin, max_zornMay 2 at 1:30

• Personal choice. There is generally no difference in meaning. – Brian May 1 at 11:41
• To distinguish $+\infty$ from $-\infty$, and from the $\infty$ denoting and element added to $\mathbb{R}$ which neighborhoods are the complements of all compact sets of $\mathbb{R}$. – logarithm May 1 at 11:42
• E.g. if you are working in extended reals $\overline{\mathbb R}=\mathbb R\cup\{-\infty,+\infty\}$ and are consistent. – drhab May 1 at 11:43

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 $$a, so if $$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?

• Okay cool, thanks. – Gurjinder May 1 at 11:47

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