Given a total ordered abstract set $(S,\preceq)$. The set is said to be complete if any non-empty subset that is bounded from above/below has a supremum/infimum.
Given a $(S,\preceq)$ may not be complete, could we always complete it in certain ways?
To be specific, I am considering the set of Hardy's L-functions which are eventually non-negative, and $f\preceq g$ iff $\lim_{x\to +\infty}\frac{f(x)}{g(x)}\leq 1$. I need to either prove this total ordered set is complete, or show that it can be complete.