Is Least Upper Bound of Empty Set equal to Greatest Lower Bound of a another Set?

Question

We had this discussion today that $\operatorname{LUB}(\varnothing) = \operatorname{GLB}(L)$ in a complete lattice $(L,\leq)$. I'm not still not getting it why that is the case.

Answer 1

The minimal element of the lattice is greater than every element in the empty set (in which there are none), and every element that is greater than every element in the empty set (which is every element) is greater than the minimal element. Therefore by definition it is the least upper bound of the empty set.

Answer 2

This is a matter of vacuous truth.

In other words, if the minimum element of $L$ is not the least upper bound of $\varnothing$, then we should find a counterexample of some element in $\varnothing$... oh wait. We can't find such counterexample! So it must be the least upper bound.