In a "standard" math yes, infinity is not a number. Nevertheless, your equations do not prove that. You can perform the same tricks with zero instead of infinity. Just replace ∞ with 0, and you can follow the same reasoning, Still zero is a number.

But sometimes it is useful to treat infinity as a number. There is nothing wrong with it. You just need to carefully define and follow the rules of its usage. And you are right, not all infinities are equal to each other.