Question about supreme of infinity and supreme of empty set Good morning, i have a few question about supreme. 1- ¿What happen with the supreme of infinity? I think not have a supreme, because you need a upper bound, but i dont know. 2- ¿What happen with supreme of empty set? Well, in this question i don't have idea how answer to this question, can you help me?

(I don't understand your first question. We usually take supremum of a set and not of an element ($\infty$ is not a number).)

The supremum of the empty set does not exist by definition:

(From Wikipedia)

> The supremum of a subset S of a partially ordered set T is **the least element in T** that is greater than or equal to all elements of S, if such an element exists.

As pointed out in the comment below, it is common to extend this definition to say that the supremum of the empty set of $=-\infty$. This might, of course, depend on the exact situation.