Definition of Greatest Lower Bound / Least Upper Bound 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!

You're right -- there is an error in the statement. Presumably, $l$ is a lower bound for $S$, so it should be true that $l\leq s$ for each $s\in S$.
