# Confusion about definition of least upper bound

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 ?

• "as I actually should" No. You should not. There are two aspects of least upper bound. The "upper bound aspect", that does target all $\alpha \in A$. ANd the least aspect which most certainly does NOT target all elements of $A$. In fact it doesn't target any elements of $A$. As lest upper bound its target are the REAL NUMBERS (nothing to do with $A$) that are all smaller than $M$. What is is saying is "$\forall w; w< M;w$ is not upper bound. Now its the target of $w$, not of $M$, that is the elments of $A$. Namely $w$ is not u.b. so $\exists a\in A; w< a$...... Sep 19, 2022 at 18:17
• The actual statement is twofold i) $\forall a\in A: a \le M$. (that is "$M$ is an upper bound") and ii) $\forall w\in \mathbb R, w<M: \exists a\in A, w< a$. Sep 19, 2022 at 18:20

• 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 $$M$$.
• I understood that. The problem that I have is with how the variable $a$ has been quantified. My problem is understanding why the variable $a$ has been quantified the way it is. Shouldn't the variable $a$ be quantified with a $\forall$ as well ? So $\forall \epsilon \in \mathbb{N}, \forall a \in A, M - \epsilon < a$ ? Sep 19, 2022 at 17:49
• No, the $\forall a$ goes only with the upper bound part. Also, note that $\epsilon >0$ (any positive real number), not $\epsilon \in \mathbb{N}$. Sep 19, 2022 at 17:51
• The least upper bound of $(0,1)$ is $1$. Why? Because for all $a\in A$ we have $a\le 1$ and for all $\epsilon >0$ the number $1-\epsilon$ is not an upper bound. Let's test that for $\epsilon =0.1$. Is $0.9$ an upper bound? No, because there exists $a\in A$ ($0.95$ is an example) such that $a \not\le 0.9$. It just isn't true (or relevant) that all elements $a \in A$ satisfy $a \not\le 0.9$. Sep 19, 2022 at 18:07