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 $f$?