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Suppose $x$ can be any real number whatsoever. Can it be said that $x < \infty$ is true for every possible $x \in \Bbb R$?

I'm asking the question in the context of infinity as defined by a left open and right unbounded interval in which $x$ is only noted to be a real number. Can it then be said that an $x$ belonging to such an interval will necessarily be smaller then infinity even though the entire real numbers set doesn't actually have a maximal number that you can point to and say that it's smaller then infinity, or perhaps can it just be said that since a right unbounded interval effectively defines its upper bound as infinity, that any $x$ from there will always be smaller then infinity by defintion?

For reference here's the wikipedia page for classification of intervals in which a left open and right unbounded interval is decipted

https://en.wikipedia.org/wiki/Interval_(mathematics)#Classification_of_intervals

Start
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    You've defined $x$ (as a real number), but you haven't defined $\infty$. – Blue Mar 27 '21 at 12:08
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    I think this link may be helpful to your qeustion – Hugo Mar 27 '21 at 12:14
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    It depends on how you define $\infty$. For example: Do you mean Cardinality of $\mathbb R$? As that is also one kind of infinity and it is "smaller" than Cardinality of power set of $\mathbb R$. Or did you mean $\infty$ as mentioned in one of the answers below? – Koro Mar 27 '21 at 12:16

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This depends on what you mean by $\infty$. If the symbol stands for projective infinity, then no. The projective real line is ordered cyclically. If you mean positive infinity of extended real line ${\overline {\mathbb {R} }}$, then yes.

Anixx
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    Why the downvote? Downvoter, what you do not agree with? – Anixx Mar 27 '21 at 12:13
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    I am not a downvoter, but the user hasn't defined infinity yet. In fact, there are three answers and not once has the OP looked at their question. I cannot blame the OP because he/she could have written this and maybe get back after some time, but until then there shouldn't have been answers. The comments above needed redressal from OP before an answer could be posted. This is not to say you are wrong, but if the OP suddenly says "my infinity is not the projective infinity" then you are in a quandary. Kindly optimize your effort to answer the right questions, it will greatly help you. – Sarvesh Ravichandran Iyer Mar 27 '21 at 13:18
  • I'm not saying that you should never answer a question without a receiving a single response from the OP. But there's a reason why the post has $-1$ votes as of writing, and that has to do with what infinity means, which hasn't been clarified. If small things aren't clarified, we are good, but the object central to the whole situation is unclear. In this version of the problem , the best would be to comment, like some people above did, and then answer the question. This will save you time and effort. – Sarvesh Ravichandran Iyer Mar 27 '21 at 13:20
  • I have edited in an explanation as to the context in which I'm referring to infinity. I hope that helps. – Start Mar 27 '21 at 14:26
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There are many different types of infinity, but in one way, this is true.

In the “extended real line,” we add to $\mathbb R,$ the real line, the two symbols $+\infty$ and $-\infty,$ and then we extend the ordering so that for every real $x$ we have $-\infty<x<+\infty.$

This is all just definition, or abstract nonsense, but it can be useful.

It does not really help much to understand the concept of the infinite.

It is important to not think of $+\infty$ and $-\infty$ as real numbers. Doing arithmetic with them is generally unwise.

Thomas Andrews
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  • I have one question here: Is it right to say that $\lim |x|=\lim |x|^2=\infty$ as $x\to \infty$? (On extended reals). Or is it nonsense to talk about this equality? (on extended real line) ? Usually this is what is said in such a situation: both the limits have the same limiting behavior. – Koro Mar 27 '21 at 12:20
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    @Koro it is also right $$\lim_{n\to \infty} n^n=\lim_{n\to\infty}n=\infty$$ – lone student Mar 27 '21 at 12:27
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    @koro Yes, there is a way you can formalize $\lim_{x\to+\infty} |x|=\lim_{x\to+\infty} |x|^2=+\infty.$ Also, I'm using $+\infty$ to be precise, but often mathematicians will just say $\infty$ rather than $+\infty.$ – Thomas Andrews Mar 27 '21 at 15:07