Totally Ordered Set successor and predecessor unique I'm trying to prove that, in a totally ordered set, an element can have at most one successor and at most one predecessor. I know that if $x < y$ and there is no $z\in X$ with $x < z < y$ then $x$ is a predecessor of $y$ and $y$ is a successor of $x$. I know that the successor and predecessor are unique but don't know how to establish it in a proof. Any advice would be greatly appreciated.

Note that the characterizations that you gave of successor and predecessor are _definitions_. In particular this means:

* $x$ is a predecessor of $y$ if _and only if_ $x * $y$ is a successor of $x$ if _and only if_ $x


So, suppose (for example) that $y_1,y_2$ are successors of $x.$ Since $y_1$ is a successor of $x,$ then $x
As similar proof approach works for uniqueness of predecessors (if they exist).