I'm reading Aubin's book "Mathematical methods of game and economic theory". On page 541, he considers a Hilbert space $V$ with a closed convex cone $P$ that satisfies $$P=\{ x \in V : (x,y) \geq 0 \quad \forall y \in P\}$$ and uses this to define an ordering by $$x \geq y\quad\iff\quad x-y \in P.$$ Then he proves the statement
Any non-empty "complete" subset $M$ bounded from above (for the ordering) has an upper bound.
I think "complete" means completely ordered. In the proof, he says
let $a \in V$ be an upper bound: $M \subset a-P$
(this I believe is equivalent to $m \leq a$ for all $m \in M$) and later on
If $z \in V$ bounds $M$ above, i.e., $z \geq m$ for all $m \in M$
So I struggle to see what the difference this and why this result isn't trivial.
Here is the text of interest.
Perhaps the author is using "the upper bound of $M$" to mean "the least upper bound of $M$"? This would explain the use of the definite article "the", and also why, as the final step of the proof, before concluding that $\overline x$ is "the" upper bound of $M,$ he shows that if $z\in V$ bounds $M$ above then $z\ge \overline{x}.$