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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'.

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  • $\begingroup$ It's not so easy, $x_n$ and $y_n$ may be incomparable for all $n$. But you can replace $y_n$ with $z_n = x_n \vee y_n$. $\endgroup$ – Daniel Fischer Jul 6 '15 at 18:28

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