Why is the supremum of the empty set $-\infty$ and the infimum $\infty$? I read in a paper on set theory that the supremum and the infimum of the empty set are defined as $\sup(\{\})=-\infty$ and $\inf(\{\})=\infty$. But intuitively I can't figure out why that is the case. Is there a reason why the supremum and infimum of the empty set are defined this way?

This is commonly done so that
$$\sup(A\cup B)=\max(\sup(A),\sup(B))$$
and
$$\inf(A\cup B)=\min(\inf(A),\inf(B))$$
In other words, as a set decreases, its supremum should decrease. The smallest a set can be is the empty set and the smallest its supremum can be is $-\infty$. As a set decreases, its infimum should increase. The smallest a set can be is the empty set and the largest its infimum can be is $\infty$.
