# Proof that $\mathbb{R}$, the set of all Dedekind Cuts, satisfies the least upper bound property.

So I just have a quick question about the beginning of this proof in a textbook.

It goes as follows:

Let $\mathbb{R}$ be the set of all dedekind cuts, and let $S \subseteq \mathbb{R}$ such that $S$ is bounded above. Define $S^* = \bigcup_{C \in S} C$

Then, $S^*$ is clearly non-empty.

Right here is where I don't understand (and the proof gives no follow-up on this claim). How do we know that $S^*$ is clearly non-empty?

Since $\emptyset \subseteq \mathbb{R}$, and $\emptyset$ is bounded above, how do we know that $S^* \neq \emptyset$? My suspicion is if $S = \emptyset$, then the "$C \in S$" part of the Union wouldn't make any sense since there are no elements in the empty set. Thus, $S^*$ would not be well-defined. However, I could be wrong, so I wish that someone could clarify more precisely why $S^*$ is non-empty. Thank you!

It's a typo. They should've written "let $S\subseteq \Bbb R$ be non-empty such that $S$ is bounded above".