Bourgain-uo sequential completeness in vector lattices

We revisit Bourgain’s 1981 counterexample to the sequential completeness of the `pointwise plus domination’ convergence on ${\mathrm{\ell }}_1$ from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain–uo convergence coincides with ordinary order convergence. Motivated by Bourgain’s construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$. We first show that sequential Buo-completeness forces $\sigma$-order completeness. Thus every non-$\sigma$-order complete vector lattice fails sequential \Buo-completeness. In particular, free Banach lattices $\mathrm{F}\mathrm{B}\mathrm{L}(E)$ are not sequentially Buo-complete whenever $\dim E>1$. On the positive side, we prove that the classical sequence lattices $c_0$ and ${\mathrm{\ell }}_{\mathrm{\infty }}$ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. Finally, we obtain a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice ${\mathrm{L}\mathrm{i}\mathrm{p}}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete.