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?

In addition to the reasons mentioned above about it making things easier, it also makes sense intuitively if you think about how each is defined. The supremum is the lowest upper bound on a set, so, since any real number is an upper bound on the empty set, no real number can be the lowest such bound (If x is that bound, then x - 1 is a lower upper bound.) Thus, it is defined to be $-\infty$ -- less than any real number, and similarly for the infimum.
