Real numbers: constructions vs. proof methods

[Timothy Y. Chow]

A recent question on MathOverflow may interest FOM readers.

https://mathoverflow.net/questions/415194

I reproduce the text of the question below.

Tim

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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?