In [this](https://mathoverflow.net/q/288105/118366) question I had asked about proof of the property of selective ultrafilter. As was answered, the proof is trivial if we know that ultrafilter is selective iff it is Ramsey ultrafilter. The proof of latter fact can be found in book "Theory of ultrafilters" by Comfort and Negrepontis. Non-trivial part of this proof is the implication "selective $\implies$ quasi-normal". While it is easy to see that the proof of latter implication is correct, I was unable to understand intuitive idea of this proof. So, I tried to find more easy and intuitively clear proof. So, the question :


Proof that for any selective ultrafilter $\mathcal{F}$ and for arbitrary family of big sets $\{A\_i\}\_{i\in\omega}$ there exists a big set $A$ with the property: $\forall i,j \in A : i<j \implies j\in A\_i$


I will give my proof in answers below.



Selective ultrafilter on $\omega$ is normal. Clear proof
I don't have my copy of Comfort & Negrepontis handy, so the following might be essentially the same as their proof, but I think it's clear enough.


The very rough idea is that, because $\mathcal F$ is a Q-point, the proof would be easy if each $A\_i$ were a final segment of $\omega$ (Steps 3 and 4 below), and, because $\mathcal F$ is a P-point, we can reduce to this final-segment case by restricting to a suitable set in $\mathcal F$ (Steps 1 and 2 below). 


Step 1. Get a big set $B$ that is almost included (i.e., included except for a finite set) in each $A\_i$. This is essentially the fact that a selective ultrafilter is a P-point, plus the equivalence of a couple of definitions of P-point. In detail: Without loss of generality (because $\mathcal F$ is non-principal) $A\_i$ has no elements $\leq i$, and so, for every natural number $x$, we can define $f(x)$ to be the smallest $i$ such that $x\notin A\_i$. This $f$ cannot be constant on any set in $\mathcal F$, because the set on which it has value $i$ is disjoint from $A\_i\in\mathcal F$. So $f$ is one-to-one on some $B\in\mathcal F$. On $B-A\_i$, this one-to-one $f$ takes only values $\leq i$, so $|B-A\_i|\leq i+1$. That completes step 1.


Step 2. Partition $\omega$ into a sequence of finite intervals $I\_0,I\_1,\dots$, where $I\_n=[e\_n,e\_{n+1})$ (with $e\_0=0$ and $e\_{n+1}>e\_n$), choosing the sequence $(e\_n)$ growing so rapidly that, for all $i\leq e\_n$, all elements of the finite set $B-A\_i$ are $<e\_{n+1}$. It is trivial to choose such $e\_n$'s by induction on $n$. This choice ensures that, if $i\leq e\_n$ and $j\geq e\_{n+1}$ (thus, if $i<j$ are in $B$ and lie in $I$-intervals with at least one other $I$-interval between them) then $j\in A\_i$.


Step 3. Apply selectivity to the function $g$ that is constant with value $n$ on exactly the interval $I\_n$, to get a set $C\in\mathcal F$ that has at most one element in any $I\_n$. So,if $i<j$ are in $B\cap C$ and are not consecutive in $C$, then $j\in A\_i$.


Step 4. Finally, let $D$ consist of every second element of $C$, and let $E=C-D$. Because $\mathcal F$ is an ultrafilter, it contains exactly one of $D$ and $E$, so it contains one of $B\cap D$ and $B\cap E$. Within either of these sets, $i<j$ implies $j\in A\_i$, so the proof is complete.


