Infimum of the empty set

Given a set $U\subset\mathbb R$ we say that $x$ a minorant of $U$ if $x<y$ for all $y\in U$, that is to say,

where $\rightarrow$ stands formaterial implication .We recall that $p\rightarrow q$ is true when the proposition $p$ is false, according to the following table:

In particular, since

for all $y\in\mathbb R$, we have that the implication

We therefore conclude that:

• every $x\in\mathbb R$ is a lower bound for the empty set.

As a consequence, the greatest lower bound for the empty set is $+\infty$: