A net in a vector lattice is said to be order convergent to if there is a decreasing net with infimum such that for any there is some such that for all .
The following lemma should exist in the literature, but we are not aware of a reference.
Proof. Suppose that is order convergent to in . There is a decreasing net with infimum in such that, for any , there is some such that for all . Clearly, 0 is a lower bound for in . Suppose that is also a lower bound. Since then for all , we have that by the definition of order ideal, and then also that . Hence , so that 0 is the infimum of in . We see that is order convergent to in . With and as above, we have that for . Hence the tail is bounded by .
Conversely, suppose that is order convergent to in and such that the absolute value of one of its tails is dominated by a positive element of . It is sufficient to prove that such a tail is order convergent to in , so we may as well suppose that for all . Let be a decreasing net in with infimum in such that for any there is some such that for all . Set , then is a decreasing net with infimum in and for all . Hence is order convergent to in . ◻