I had a question about the Least Upper Bound Property. So it states that every non-empty subset of $\mathbb{R}$ that has an upper bound must have a least upper bound in the reals.
My question is: Does the set of all real irrational numbers with simple order have the least upper bound property?
I was thinking about this and said no. This set isn't necessarily bounded (at least I don't see how it's bounded) which is why I don't think it has the least upper bound property. But someone was telling me this set has the least upper bound property and their reason was it's a subset of the reals and every subset of the reals has a least upper bound property.
So I'm not sure if I'm right or my friend is right, but I feel like they're wrong which is why I was wondering.