Im reading a proof and I get lost in a phrase. The statement to be proved:
Let $X$ a topological space and $G_0$ a collection of subsets of $X$ with the finite intersection property. Then there exists a maximal collection $G$ of subsets of $X$ with the finite intersection property that contains $G_0$.
Now the proof starts like this:
The family of all collections of sets with the finite intersection property and containing $G_0$ is partially ordered by inclusion, so that by the Hausdorff principle there is a maximal linearly ordered subfamily $F$. We claim that $G$ is the union of all the collections in $F$.
Any $n$-tuple $\{E_1 ,...,E_n \}$ of elements of $G$ belongs to at most $n$ collections $G_j$. Since $\{G_j\}$ is linearly ordered there is a collection $G_n$ that contains the others.
...and so on.
But I can't follow the emphasized phrase, indeed I have a lot of counterexamples: by example let the collection $G_0:=\{(0,1),\,(0,2),\, (0,3)\}$ of subsets of $\Bbb R $. Then $G_0$ have the finite intersection property and a chain of collections that contains $G_0$ can be defined recursively by $G_n:=G_{n-1}\cup (0,3+n)$. Now let $G:=\bigcup_{n\geqslant 0}G_n$, and so the $3$-tuple defined by $G_0$ belongs to every $G_n$ for all $n\in \Bbb N $, not just to at most three collections of the chain.
Maybe Im misunderstanding the emphasized phrase, due to a lack of knowledge of the english language? Or maybe it is just a typographic error and it must say "at least" instead of "at most", what seems to have more sense in a linearly ordered chain. Some help will be appreciated, thank you.
EDIT: you can read the original text of the proof here.
I concur with @freakish' comment. The author means that if we have a maximal chain $\mathcal{F}$ in the poset with union $\mathcal{G}$, and any set $\{E_1,E_2,\ldots, E_n\}$ (it's not a tuple) that is a subset of $\mathcal{G}$, we can find (at most) $n$ families $\mathcal{G_j}\in \mathcal{F}, j \le n$ such that $E_i \in \mathcal{G}_i$ for all $i$, just by the definition of a union, and as the set of families $\mathcal{F}$ is linearly ordered by inclusion, one of them $\mathcal{G}_{j_0}$ say, is the largest and so all $E_i$ lies in that one family (that has the FIP by virtue of being in the poset at all) and so $\bigcap_{i=1}^n E_i$ is non-empty, etc.
It's a special case of the argument that for a property of finite character we can always has a maximal family with that property, a special case of Zorn (also used in the proof of a basis in a vector space, and in a proof of the Alexander subbase lemma etc.). This is known as the Teichmüller-Tukey lemma.