Prove that $|\mathbb{R}| = |(0, 1)|$. Prove that $|\mathbb{R}| = |(0, 1)|$. (Hint: Consider the tangent function.)
This is my current thought process:

Using the hint, I map $(0, 1) \rightarrow (-\frac{\pi}{2},\frac{\pi}{2})$ by the function $f(x) = \pi x - \frac{\pi}{2}$, and
  then state that since $f$ is linear, and bijective, it must be that -- somehow --
  $|\mathbb{R}| = |(0, 1)|$.

Do I have the general idea? Or am I way off?

Follow the hint and use the tangent function. The definition of two sets having equivalent cardinality is that there exists a bijection between them.
