TITLE: Connectedness and the real line
QUESTION [7 upvotes]: It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be seen in Munkres's topology or Lee's Introduction to topological manifolds)
This seems strange to me, because for other fundamental results such as the Compactness of $[0,1]$, I can think of several proofs. 
Does anyone know any different proofs of the connectedness of $\mathbb{R}$?

REPLY [2 votes]: If $\mathbb{R} = A \uplus B$, for $A,B$ nonempty open subsets, then each of them is a countable union of disjoint open intervals (to prove this, one has to use the completeness of $\mathbb{R}$). Let $(a_1,a_2)$ be an interval which appears in this decomposition of $A$. Then $a_2$ can't be in $A$, but also can't be in $B$ since $B$ is open. Contradiction.