$|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval I want to prove that any open interval $(a,b)$ has the same cardinality of the real numbers: $|(a,b)| = |\Bbb R|$. 
Do I have to find an function to prove it? Or is there a theorem to prove it easier? or any idea?
 A: The function $y = \tan(x)$ is bijective on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. We would like to shift/stretch it so that it's bijective on $(a,b)$. The period should be $b-a$, so we would at least have $y = \tan\left(\frac{\pi}{b-a}x \right)$. Now translate it.
A: Hint: First find an easy bijection between any two given open intervals (you can even find a continuous one), and then use a function and certain interval that you can show a bijection to ${\mathbb R}$ with (for example, tangent function).
A: There's a great visual proof for this, have you seen it?
It's simple enough that I can just describe it, and perhaps you can draw it yourself:
Take a circle $C$ of circumference $|a-b|$ (this will be our interval $(a,b)$, after we remove a single point), and place it above a straight line $L$ of infinite length having no endpoints (this is our stand-in for $\mathbb{R}$). Mark the point $v$ on $C$ that is furthest from $L$. 
The bijection between the points on $C$ (excluding $v$) and the points on $L$ is simply: the point $u$ on $C$ maps to the point $p$ on $L$ if both lie on a line segment from $v$ to $L$.
