I have read the proof that the infinite sets \(\displaystyle |(0,1)| = |\mathbb{R}|\).

We know that \(\displaystyle (0,1) \subset \mathbb{R}\), and I know there is a bijective function between the two sets, but how does one explain that \(\displaystyle (0,1) \) and \(\displaystyle \mathbb{R}\) have the same number of elements?