I would be grateful if one could confirm that the following argumentation is fine.
Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive elements $x,y\in \overline{\bigcup_n L_n}$ with $x\leqslant y$, can we find two sequences $(x_n)$ in $(y_n)$ in $\bigcup_n L_n$ such that
- $0\leqslant x_n \leqslant y_n$
- $x_n\to x$ and $y_n\to y$ as $n\to \infty$?
I guess so. Since $x$ and $y$ are positive, they are limits of positive sequences in $\bigcup_n L_n$. (Is there any reference for that?) Thus, for almost all $n$ we must have $x_n\leqslant y_n$ as otherwise $x\geqslant y$. So we simply delete from our sequences fintely many `bad pairs'.