If is any well-ordered set of real numbers, then every subset of has a least element. If is a nonempty subset of , then has a least element.
Question: Since is well ordered and has a least element, and since having a least element is not a sufficient condition for a nonempty set to be well-ordered, does it mean that is not necessarily well-ordered?
I honestly have spent the entire time, since your last reply, thinking of the question you put forth. I think I know what you were trying to show me, but I want to know if I understood it well.
Here is the answer:
Since the set of real numbers is a well-ordered set, by definition, every subset of must have a least element, and since is a subset of A, it follows that every subset of is a subset of containing a least element. In turn, every subset of has a least element. Consequently, by the same definition, is a well-ordered set.
I think you can be proud of me.