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.