Every open interval (a,b) with a < b is uncountable in $\mathbb{R}$ So I want to show that a open interval ($a$,$b$) with 
$a$ $<$ $b$ is uncountable in $\mathbb{R}$.
My idea:
we can show that:
For $a$ ,$b$ , $c$ $\in$ $\mathbb{R}$ the interval ($a$,$b$) is equipotent to ($a$+$c$,$b$+$c$). We can use this fact for the proof :
We know that $\mathbb{R}$ is uncountable. Maybe we could $\mathbb{R}$ write as a countable union of intervals, which are equipotent to ($a$,$b$) ? 

The map $f:(a,b)\rightarrow{\bf{R}}$, $f(x)=\tan\left(\dfrac{\pi}{b-a}\left(x-\dfrac{a+b}{2}\right)\right)$ is a bijection.
