Proving the uncountability of $[a,b]$ and $(a,b)$ I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof  involves perfect sets.
Can anyone please point me to a proof of the above facts without point-set topology?
I am thinking along these lines:
$\mathbb{R}$ is uncountable.If we can show that there exists a bijection between $(a,b)$ ad $\mathbb{R}$ we can prove $(a,b)$ is uncountable.But I am not sure how to construct such a bijection. 

$$\tan \left( \frac{\pi}{(b-a)} (x-\frac{a+b}{2})\right)$$
Basically $f(x)=\frac{\pi}{(b-a)} (x-\frac{a+b}{2})$ is the linear function such that $f(a)=-\frac{\pi}{2}$ and $f(b)=\frac{\pi}{2}$.
