Order in a quotient space of $\mathbb{R}^\mathbb{N}$ ($\mathbb{R}^\omega $)

Question

Let $\mathcal{F}$ be a filter in $\mathbb{N}$ finer than Fréchet filter.

In $\mathbb{R}^\mathbb{N}$ we define the equivalente relation : $(a_n) \equiv (b_n)$ iff $\{n | a_n = b_n\} \in \mathcal{F}$. Prove that is consistent with the order in the sequences.

I think that I have to use the dictionary order but I'm not sure. Also, I think the goal is to prove that if $(a_n) \leq (b_n)$ then, if $(a_n) \equiv (c_n)$ and $(b_n) \equiv (d_n)$ implies that $(c_n) \leq (d_n)$.

Thaks for the help

Answer 1

Let $A=\{n:a_n\leq b_n\},\;$ $ B=\{n:a_n=c_n\},\;$ $ C=\{n:b_n= d_n\},\;$ $D=\{n:c_n\leq d_n\}.$ $$\text { We have }\quad [\;(a_n)_n\leq (b_n)_n \land (a_n)_n\equiv (c_n)_n\land (b_n)_n\equiv (d_n)_n\;] \iff$$ $$\iff (A,B,C\in \mathcal F)\implies A\cap B \cap C\in \mathcal F\implies$$ $$\implies D\supset \{n:c_n=a_n\land a_n\leq b_n \land b_n=d_n\}=B\cap A \cap C \in \mathcal F\implies$$ $$\implies D\in \mathcal F\iff (c_n)_n\leq (d_n)_n.$$