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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 $[a,\infty)$?

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

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    $\begingroup$ Personal choice. There is generally no difference in meaning. $\endgroup$ – Brian May 1 at 11:41
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    $\begingroup$ 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 $\mathbb{R}$. $\endgroup$ – logarithm May 1 at 11:42
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    $\begingroup$ E.g. if you are working in extended reals $\overline{\mathbb R}=\mathbb R\cup\{-\infty,+\infty\}$ and are consistent. $\endgroup$ – drhab May 1 at 11:43
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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?

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  • $\begingroup$ Okay cool, thanks. $\endgroup$ – Gurjinder May 1 at 11:47
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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

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