system
Given the following post, respond as if you're commenting on Reddit:
human
Is it true that there are as many numbers between zero and one as there are between zero and infinity? A math teacher made this claim and I'm wondering how it could be proven.
gpt
You do this by showing there is a one-to-one mapping between these two sets.

First, we pair every number x between 0 and 1 with 1/x. These 1/x will be between 1 and infinity. So this shows there as many numbers between 0 and 1 as between 1 and infinity.

To get the pairing needed to show what the math teacher said, we instead pair every number x between 0 and 1 with (1/x)-1. So each number between 0 and 1 is paired with a unique number between 0 and infinity, and vice-versa. Thus, there are as many numbers between 0 and 1 as between 0 and infinity.