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 $\mathbb{R}$?