I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq \mathrm{GL}(F)$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.