I would like to know how to demonstrate that the real numbers and the interval [-1, 1] are equipotent
Thank you
Using that $\arctan$ function it is easy to show that $\left( { - 1,1} \right) \approx \mathbb{R}$.
Hence your question comes down to showing that $\left[ { - 1,1} \right] \approx \left( { - 1,1} \right)$
The standard proof for the latter is to construct a shift of a countable set.