The standard proof given for the connectedness of $\mathbb{R}$ seems to me to be along the lines of Understanding the proof of "connected set is interval.".
A much simpler argument, I believe, is that $\mathbb{R}$ is pathconnected and hence connected ( open subsets of normed vector spaces are connected iff they are path-connected).
Why is the linked proof prefered ( or even stated) to this argument? Is my argument false/ less general? Many thanks.
Best Answer
The fact that path-connected implies connected, in fact relies on the fact that $[0,1]$ is connected. And if you already know $[0,1]$ is connected you have already shown that $\Bbb R$ is connected too (it follows from general facts in ordered spaces; usual proofs use that bounded sets have sups etc., and proofs apply to both $[0,1]$ and $\Bbb R$ or any interval).
So proving $\Bbb R$ connected via path-connectedness is a circular argument.