To me it seems trivially true that if $A$ is a set ordered by relation $C$ denoted $<_C$ and $A$ has the greatest lower bound property (or least upper bound property), then $A$ ordered by the converse relation $D$ defined as $(b,a)\in D$ if $(a,b)\in C$ and denoted $<_D$ has the least upper bound property (or greatest lower bound property).
This isn't an individual problem I'm trying to solve, just part of an argument I'm trying to give to a problem (g.l.b. property $\implies$ l.u.b. property) in Munkres' Topology.
A proof would go something like showing how under $<_C$, given a subset $A_0\subset A$ bounded below, $A_0$ is bounded above under $<_D$. Then $\inf A_0$ under $<_C$ is such that $\inf A_0\leq_C x$ for all $x\in A\iff\inf A_0\ge_D x$ for all $x\in A_0$. And so on, eventually showing $\inf A_0$ under $<_C$ equals $\sup A_0$ under $<_D$.
Is this right?