https://www.desmos.com/calculator/powhaqmvvp
The animation shows that numbers on the x-axis, can be matched with numbers on the interval -1<x<1 in such a away that:
1. every x on the x-axis is matched with a different number in the interval -1<x<1,
and
2. all the numbers in -1<x<1 are used
hence, we can see that there must be the same number of numbers on the x-axis as there are on the interval -1<x<1.
These numbers sets have the same Cardinality.