Total Orders and Minimum/Maximum elements How can I prove that for any given Poset $(A,\preceq)$, $\preceq$ is a total order implies that $\forall a\in\preceq$, if a is a maximal, then a is maximum? Same goes for minimal/minimum.

I assume that you are asking the following:
In a totally ordered set, why is every maximal element a greatest element?
The answer is simple: Assume $a$ is maximal. Let $b$ be an arbitrary element of our set. Since the set is totally ordered set we have either $a\leq b$ or $b\leq a$. Since $a$ is maximal, we must have $b\leq a$. Since $b$ was arbitrary, it follows that $a$ is a greatest element.
