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Consider the following two expressions: $$\sum^{\infty}_{i=1}\frac{1}{i}$$ and $$\lim_{h\to90 h<90}\tan 90º$$ They both equal to infinity.

I remember my teacher told me there are more real numbers than whole numbers. So $\infty>\infty$ is possible.

But how do I know if two expressions that both equal to infinity are equal or not (assume there isn't an obvious bijection between them)?

Thank you in advance.

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    $\begingroup$ No, "$\infty>\infty$" makes no sense. And there are not "more rational numbers than integers". The rational numbers can still be counted. But there are "more real numbers than integers" because the set of reals is uncountable. $\endgroup$ – Peter Jun 8 '18 at 7:14
  • $\begingroup$ @Peter But there are infinity many real numbers and infinity many integers. $\endgroup$ – abc... Jun 8 '18 at 7:21
  • $\begingroup$ Yes, but there is no bijection between them. Cantor showed this with his diagonal argument. This is an important result in set theory. $\endgroup$ – Peter Jun 8 '18 at 7:23
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I think you're confusing two different concepts related to infinity - cardinality and limits.

Your teacher probably taught you that there are "more" real numbers than rational numbers (that's not the case for whole and rational numbers, by the way) because both infinite sets have different cardinality. Meaning, you can't make a one-to-one correspondence between the reals and the rationals, so one might say that the infinity of the real numbers is "bigger" than the infinity of the rational numbers. Infinite cardinals are about functions that map from one infinite set to another.

But the meaning of an infinite limit is different, and isn't really related to the concept of cardinality. When we say that the limit of a sequence is infinity, we mean that for any number you choose, starting some point, all members of the sequence are greater than that number. To put it simply, it means that the sequence gets bigger and bigger, tending "towards" infinity.

The two terms relate to infinity in different ways - the first talks about how infinite sets relate to one another, and the second talks about how sequences and functions behave under certain conditions. That said, there's no such thing as the cardinality of a limit, or the limit of a cardinality, so you can't really compare the two.

If you find this topic interesting, I'd recommend you to start reading some basic texts on elementary set theory and real analysis. I think it might help you get a deeper understanding of these ideas.

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  • $\begingroup$ Superb answer! (+1) $\endgroup$ – Peter Jun 8 '18 at 9:47

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