$\mathbb R $ is uncountable The uncountability of $\mathbb R $ can be proved by two
beautiful methods..
One is by proving the sequence of 0and 1 are uncountable using Cantor's diagonal process in which we choose any countable subset of the set of all  sequence of 0 and 1. And then by altering the i'th components of i'th element of the countable subset we get a sequence of 0 and 1 which lie outside the countable set..And considering the binary representation of all reals. 
The another method ,as described in munkres' Topology ,is to prove any nonempty compact Hausdorff space with no isolated point is uncountable. This would prove intervals of $\mathbb R $ is uncountable and hence $\mathbb R $  is uncountable.
Both the methods are beautiful but is there any relation between the two?? Two argument proving the same thing has no Linc at all, is it possible?
 A: The uncountability of the reals can be established in many more ways. The proofs can be divided into two classes. One, like Cantor's proof you mention, relies on some syntactic way to represent real numbers (in Cantor's case it is the essential uniqueness of decimal expansions, so essentially proving that countably long sequences of at least two symbols is an uncountable set, which is a consequence of the ability to manipulate such sequences in a certain way, and then making a connection between the reals and such sequences). The other way, as in the one you mention from Munkres, does not rely on any particular way of representing the reals. It is agnostic to how you express the reals. It only relies on properties of the real number system, not of the way the numbers are written down. 
Ultimately, the notation must reflect somehow the properties that make the proof work, but these are really two very different styles of proof. It should not be surprising that these proofs, even though they prove the same result, seem unrelated. They take entirely different approaches. This example shows us the beautiful interplay between notation and properties. A good notation yields sometimes very simple proofs of deep results. 
