There is indeed an higher order of infinity for the real numbers between 0 and 1 than integer between 0 and infinity.

The proof (simplified) goes as follow:

For each integer between 0 and infinity (let’s call that number x) you can match it with a number between 0 and 1 that contains a number of 1 after the dot equal to our integer x. So you get:

1 -> 0.1

2->0.11

3->0.111

4->0.1111

etc.

As you can see, going to infinity, we will have two matched sets where every element is different. However, if we add a 2 just after the dot in the real number set, we get 0.21, 0.211, etc, an entirely NEW set, of which no elements were contained in any of our previous sets. There is actually an infinite number of these transformations that can be made to the real set, and none that can be made to the integer set.

The integer set is named a countable infinity, where although there is an unlimited number of element, if you choose two different elements, there is a finite number of elements between them.

The real numbers are an uncountable infinity, where if you chose any two elements, there is still an infinite number of elements between them.