Math Help - introduction to real analysis

1. introduction to real analysis

I would like to know how to demonstrate that the real numbers and the interval [-1, 1] are equipotent

Thank you

2. Re: introduction to real analysis

Originally Posted by Yeison
I would like to know how to demonstrate that the real numbers and the interval [-1, 1] are equipotent.
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.