Only if, when they say from 1 to infinity they mean integers. There are countably infinite integers, where there are uncountably infinite real numbers.

There's less Integers between 1 and "infinity" than there are Real numbers between 0 and 1. But there are equally many real numbers between 0 and 1 as there are real numbers between 1 and "infinity"

If I remember correctly, it has to do with rational numbers from one to infinity being countable infinite and the real numbers between 0 and 1 being uncountable infinite. The trick to the proof comes with being able to count rational numbers from smallest to largest (easy to think about with integers but even with rational numbers it's just integers in the numerator and integers in the denominator so just assign a count to the numerator first and the next count to the denomator and it works). for real numbers if you try to count from one number to the next, there will ALWAYS be an number in between those that you missed and should have counted. blew my brain when the professor showed the proof in class.

lemme try.

for every x in R such that 1 < x < infty, there is exactly one number (1/x) in R, s.t. 0 < (1/x) < 1.

also vice versa.

there's a bijection between the two sets, therefore they are the same size.

am i missing something?

No. The cardinality of any real interval, bounded or unbounded, is the same. You can find a bijection between any of them.

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.

Yeah I should’ve said 1 to infinity! Title typo

cantor's diagonal