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. 

If (a, b) is countable, then we have a bijection between Q and (a, b).  Q bijects with any infinite set of rational numbers (c, d).  Thus, we can encode all points of (a, b) by a countably infinite list of all rational numbers as follows:
1-x.kfj...
2-y.fli...
3-i.opq...
.
.
.
Find the first non-zero digit of "a" in its infinite decimal expansion.  Put 0s down on paper until you reach this digit position.  Then choose the numeral from the next digit position from rational number 1 and use it basically as follows... if we have a=.00700... and the first rational number on our list is 1.234000..., then we choose 0, since 7 comes in the third position after the decimal point, and 0 lies in the fourth position.  So, our number so far is .0000.  Then choose the numeral in the next digit position from rational number 2, suppose it's .00005454...  Then we have .00005, and so on.  We end up with a number greater than "a", yet less than "b", nowhere on our list of rational numbers.  Thus, there exists no bijection from Q to (a, b), and thus (a, b) is uncountable.  Since (a, b) is uncountable, [a, b] is uncountable also.
