Honestly, I figured out the idea to prove this theorem without much difficulty. But I found very hard to formalize it into a rigorous proof and it takes me three days to do it.
Could you please have a check on my attempt? Thank you so much!
A sequence $\langle a_n \rangle_{n=0}^{\infty}$ of rational numbers is a Cauchy sequence if for every rational $\epsilon>0$ there exists an integer $N$ such that $|a_m – a_n| < \epsilon$ for all $m,n > N$. Let $\mathfrak{C}$ be the set of all Cauchy sequences of rational numbers. We define a relation $\preccurlyeq$ on $\mathfrak{C}$ by $\langle a_n \rangle_{n=0}^{\infty} \preccurlyeq \langle b_n \rangle_{n=0}^{\infty}$ if and only if for every rational $\epsilon>0$ there exists an integer $N$ such that $a_n – b_n < \epsilon$ for all $n > N$.
Theorem: Any two elements of $\mathfrak{C}$ are comparable w.r.t $\preccurlyeq$.
My attempt:
Assume the contrary that there exist $\langle a_n \rangle_{n=0}^{\infty}$, $\langle b_n \rangle_{n=0}^{\infty} \in \mathfrak{C}$ that are incomparable. Then $\langle a_n \rangle_{n=0}^{\infty} \not \preccurlyeq \langle b_n \rangle_{n=0}^{\infty}$ and $\langle b_n \rangle_{n=0}^{\infty} \not \preccurlyeq \langle a_n \rangle_{n=0}^{\infty}$. As a result, $\exists \epsilon >0,\forall N,\exists n>N:a_n – b_n \ge \epsilon$ and $\exists \epsilon >0,\forall N,\exists m>N:b_m – a_m \ge \epsilon$.
Then $\exists \epsilon >0,\forall N,\exists (n>N \wedge m>N):a_n – b_n \ge \epsilon \wedge b_m – a_m \ge \epsilon$. By Axiom of Choice, there exist functions $f:\Bbb N \to \Bbb N$ and $h:\Bbb N \to \Bbb N$ such that $f(N)>N$, $h(N)>N$, $a_{f(N)} – b_{f(N)} \ge \epsilon$, and $b_{h(N)} – a_{h(N)} \ge \epsilon$.
Moreover, $\langle b_n \rangle_{n=0}^{\infty}$ is a Cauchy sequence $\implies$ $\exists N_0,\forall m,n > N_0:|b_m – b_n| < \epsilon$.
We define functions $f':\Bbb N \to \Bbb N$ and $h':\Bbb N \to \Bbb N$ by $f'(N)=f(N_0)$, $h'(N)=h(N_0)$ for all $N<N_0$ and $f'(N)=f(N)$, $h'(N)=h(N)$ for all $N \ge N_0$.
It follows that $f'(N)>N$, $h'(N)>N$, $a_{f'(N)} – b_{f'(N)} \ge \epsilon$, $b_{h'(N)} – a_{h'(N)} \ge \epsilon$, and $|b_{h'(N)} – b_{f'(N)}| < \epsilon$ for all $N \in \Bbb N$. As a result,
$\begin{align}|a_{f’(N)} – a_{h’(N)}| &= |(a_{f’(N)} – b_{f’(N)}) + (b_{h’(N)} – a_{h’(N)} ) + (b_{f’(N)} – b_{h’(N)})|\\ &\ge |(a_{f’(N)} – b_{f’(N)}) + (b_{h’(N)} – a_{h’(N)} )| – |b_{f’(N)} – b_{h’(N)}|\\ &= |a_{f’(N)} – b_{f’(N)}| + |b_{h’(N)} – a_{h’(N)} | – |b_{f’(N)} – b_{h’(N)}|\\ &> \epsilon + \epsilon – \epsilon =\epsilon\end{align}$
Hence $\exists \epsilon>0, \forall N, \exists(f’(N)>N \wedge h’(N)>N):a_{f’(N)} – a_{h’(N)}>\epsilon$. This contradicts the fact that $\langle a_n \rangle_{n=0}^{\infty}$ is a Cauchy sequence.
Best Answer
I see only a (minor) problem with your attempt: at the first paragraph, those $\varepsilon$'s should be two distinct numbers a priori. But that's not a serious problem.
On the other hand, the axiom of choice is not needed for the proof, but using it is not an error.