I have recently went through the definition of the Greatest Lower Bound (GLB) and Least Upper Bound (LUB) and I was somewhat confused with regards to the definition of the GLB in this book of mine.
For any Greatest Lower Bound of $S$, $m$:
$$glb(S)=m\iff [(\forall s\in S)(s\le l)]\to(m\ge l)$$
For any Least Upper Bound of $S$, $M$:
$$lub(S)=M\iff[(\forall s\in S)(s\le L)]\to(M\le L)$$
My question on the GLB is why $(s\le l)$ instead of $(s\ge l)$.
I mean, if $(s\le l)$, it would not seem to make sense to me as $l$ will be greater than every $s$ in the set $S$, which essentially tells me that $l$ is located in the upper bound of the set $S$.
Furthermore, the consequent $(m\ge l)$ would further mean that $m$ is located at the upper bounds of $S$, resulting in a contradiction as the definition of $m$ is the Greatest Lower Bound of $S$.
Would appreciate it lots if someone can solve this issue for me on if the definition for GLB is correct or not, thank you very much for the help!