Confusion about definition of least upper bound

Question

I'm learning set theory and I got to bounded sets. I got to the following definition and I have a question about it:

The problem that I have is the quantified variable $a$. If I would write the quantified statement down it would look like this: $\forall \epsilon \in \mathbb{N}, \exists a \in A, M - \epsilon < a$. My question is wouldn't the right quantified variable $a$ be a $\forall$ as well, like this: $\forall \epsilon \in \mathbb{N}, \forall a \in A, M - \epsilon < a$. Now, with the definition from the book, I'm not fully targetting all of the elements of $A$, as I actually should since a least lower bound is targetting the whole set. So the question is, which version is right, mine or the one from the book ?

Answer 1

There are two parts to the definition:

• Upper bound: $\forall a\in A, a \le M$
• Least: $\forall \epsilon>0, \exists a \in A, M - \epsilon < a$

For the "least" part, it might help to think of an equivalent statement: $\forall \epsilon>0, \lnot(\forall a \in A, M - \epsilon \ge a)$, which says that $M - \epsilon$ is not an upper bound. That is, there is no upper bound smaller than $M$.