In my elementary real analysis course three years ago, I remember noting that there seemed to be 3 main ways of proving the main theorems about continuity. There was Bolzano-Weierstrass, continuous induction, and the bisection argument. At the time, I explained this to myself by thinking these must correspond to the three common constructions of the reals, i.e., Cauchy sequence construction, Dedekind cut construction and (binary) decimal construction, respectively. And that they're natural ways to reason within the corresponding construction.
I was wondering if there's some sort of logical connection here, between the construction of the reals and the methods you can use for proving properties of it? I was wondering if there are other proofs corresponding to more exotic constructions such as Eudoxus reals? Is this an example of something more general?