# greater and smaller infinity?

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)?

• 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. – Peter Jun 8 '18 at 7:14
• @Peter But there are infinity many real numbers and infinity many integers. – abc... Jun 8 '18 at 7:21
• Yes, but there is no bijection between them. Cantor showed this with his diagonal argument. This is an important result in set theory. – Peter Jun 8 '18 at 7:23