An issue in constructing the real numbers using Cauchy sequences in the rationals.

Question

I'm having trouble justifying why we can discuss Cauchy sequences before the real numbers are constructed. As we know, the definition of a Cauchy sequence (in the metric space $\mathbb{Q}$) starts with "For every $\epsilon > 0 \dots$," which is where my issue lies. Suppose we use an irrational $\epsilon > 0$ that hasn't been constructed yet. How do we know that the equivalence class of Cauchy sequences representing $\epsilon$ satisfies the criterion for being Cauchy with this value of $\epsilon$ if it hasn't been defined yet?

Answer 1

You define a Cauchy series of rationals only for rational $\epsilon$. It doesn't change the definition. You could define Cauchy sequence by restricting $\epsilon$ to be of th form $1/n$ for some integer $n$.

Answer 2

For the definition of convergence, and of Cauchy sequences, it suffices to only consider rational $\epsilon$, or even only $\epsilon$'s of the form $\frac{1}{n}$, $n\in \mathbb N$. So you don't need to already have the reals constructed just to define limits/Cauchy sequences.