Max Operation Yields Supremum of Parameters

Theorem
Let $\struct {S, \preceq}$ be a totally ordered set.

Let $x, y \in S$.

Then:
 * $\max \set {x, y} = \sup \set {x, y}$

where:
 * $\max$ denotes the max operation
 * $\sup$ denotes the supremum.

Proof
As $\struct {S, \preceq}$ be a totally ordered set, all elements of $S$ are comparable by $\preceq$.

Therefore there are two cases to consider:

Case 1: $x \preceq y$
In this case:
 * $\max \set {x, y} = y = \sup \set {x, y}$

Case 2: $y \preceq x$
In this case:
 * $\max \set {x, y} = x = \sup \set {x, y}$

In either case, the result holds.

Comment
Note that it is considered abuse of notation to write
 * $\max = \sup$

This is because:
 * $\max: S \times S \to S$

while:
 * $\sup: \powerset S \to S$

where $\powerset S$ is the power set of $S$.

Also see

 * Min Operation Yields Infimum of Parameters