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.

We say that a set $A$ admits a maximal element $a$ with respect to a binary relation $\succeq$ on $A$ if there exists no element $b\in A$ such that $b\succ a$. 
We want to show that $a\succeq b$ for all $b$ in $A$. Since $\succeq$ total either $a\succeq b$ or $b\succeq a$. Let $b\in A$ be such that $b\succeq a$ since $a$ is maximal not $b\succ a$ therefore whenever $b\succeq a$ a it implies that $b=a$. Hence $a$ is maximum.
