1. How do you even express the idea of continuity if not through limits? (Sure, there's the topological definition and the epsilon-delta criterion but those are merely an abstraction or a different way of phrasing the same idea.)
2. The key difference between the rationals and the reals is that the latter are Cauchy-complete. How do you even define (in an intuitive way) what that means without referring to limits?
3. The reals are typically constructed as (equivalence classes of) Cauchy sequences, i.e. by their very definition they are the limits of their Cauchy sequences.
i think, for the purpose of introducing the continuity of the reals, for the purpose of introducing limits, one could skip any discussion that involves constructing the rationals or building the reals from them, at least to start.
instead go directly from integers to reals by introducing countable vs. uncountable infinities.
1. How do you even express the idea of continuity if not through limits? (Sure, there's the topological definition and the epsilon-delta criterion but those are merely an abstraction or a different way of phrasing the same idea.)
2. The key difference between the rationals and the reals is that the latter are Cauchy-complete. How do you even define (in an intuitive way) what that means without referring to limits?
3. The reals are typically constructed as (equivalence classes of) Cauchy sequences, i.e. by their very definition they are the limits of their Cauchy sequences.