It's a notoriously thorny matter to decide whether two proofs of a given theorem are "really different". But...a proof of the connectedness of the real line using **real induction** is given in Theorem 9 of this note of mine. This proof (to me) *feels* moderately different from the usual LUB proof, and I think I like it a little better.

Comments:

1) Actually what is proved is that any closed, bounded interval $[a,b]$ is connected. But you can get from here to the connectedness of $\mathbb{R}$ with no trouble at all: e.g. the union of a chain of connected subspaces is connected.

2) I certainly do not mean to suggest that I am the first person to prove the result in this way. On the contrary, please see the end of the paper and the bibliography for remarks about the (many) others who have argued (sometimes very) similarly.

3) Also Section 4 on "Topological Equivalents of Completeness in Ordered Sets" seems relevant to the spirit of the question. Again, there is no new result here but the issues are discussed with more thoroughness than in any one source I know. (As usual, please do interpret this as an invitation to expand my knowledge...)