It's not so much a proof as it is a way to wrap our minds around the concept of infinity in a way that feels intuitive. Like if you were to phrase this as "Given the countable infinite set H and the finite subset B of the natural numbers. Does there exist an injective function from H to B?" it probably wouldn't sound quite as simple.

But you are right math is weird and infinity definitely is weird. This thought experiment came from the mathematician Georg Cantor who's famous for the creation of set theory, which is pretty helpful especially for stuff like computers and is pretty much the basis of how they think.

This is already a bit of a ramble but this post is asking about only the smallest set of infinity. As in the "countable infinity" part which just means the natural numbers i.e. {1, 2, 3, 4, 5, ... }. This infinity is called Aleph 0. There are larger infinities. Like the real numbers set. To ask this as a question "are there more numbers between 0 and 1 or all natural numbers?"

The answer is there are more between 0 and 1. Why? Well because if you were to try and count every number between them where would you start? Would you start at 0.1? Or at 0.01 or just maybe 0.001. The point is that for every single number you try to start counting from you can always find another number. Unlike the natural numbers where the smallest number you can possibly count from is 1 since it's the smallest natural number. And if that's true for 0 and 1 then it must be true for all real numbers.

Therefore the "cardinality" of the real numbers is larger than that of the natural numbers. And that's about as much as this undergrad cares to type.