"the existence of all suprema ... equivalent to the existence of all infima. Indeed, for any subset $X$ of a poset, one can consider its set of lower bounds $B$. The supremum of $B$ is then equal to the infimum of $X$"
My question is, can $B$ be the empty set?
My idea is that $B$ cannot be the empty. This is because if we consider the empty set, then all elements of the poset is an upper bound of the empty set. But the poset has no lower bound because $X$ has no lower bound, so there is no least upper bound on the empty set, contradicting the existence of all suprema. Is this argument sound?