I read in one book: The following statements about an upper bound $u$ of a set $S$ are equivalent:
- if $v$ is any upper bound of $S$, then u $\leq$ v.
- if $z < u$, then $z$ is not an upper bound of $S$.
- if $z < u$, then $\exists \ s_z \in S$ such that $z < s_z$.
- if $\epsilon > 0$, then $\exists \ s_\epsilon \in S$ such that $u-\epsilon<s_\epsilon$.
The two lasts are ok. But the first two statements are wrong, aren't they? For example, we consider the set $A:=\{1,2,3,4,5\}$. Then $8=u$ is upper bounds because $\forall x \in A$ we have that $x\leq8=u$. Then we have by first statement that if $v$ is any upper bound of $S$ then $u\leq v$, but $v=7$ is also upper bound of $A$ and we have that $v\leq u$. And it's a contradiction to 1 and 2 statements. To the first one because $v$ is also upper bound of $A$ and to the second one because if $z<u$ then $z$ is not an upper bound of $S$ but in my example 7 was an upper bound of $A$. Where do I have something wrong? Thank you!
