Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N}\to S$. Now, I consider (0,1). I want to prove by contradiction that $(0,1)$ is not countable.
First, I assume the contrary that there exists a bijection $f$, and I can find an element in $S$, but not in the range of $f$. But I can't find such element. How can you construct such $f$?