First question I have is how would you describe the relationship between an ordered field and an ordered set and continue the proof by treating the field as a set? I want to say that right in the beginning to make things clear but I don't know how to say it.
Here's the rest of the proof:
Let A be a nonempty subset of S s.t. it is bounded above and D be the set of all lower bounds for A.
Fix a to be elements of A. Then for each d in subset D d is a lower bound for A so d $\leq$ a for all a in A.
Since d is bounded above by A it has the supremum $\gamma$ . Since every a is an upper bound for d, $\gamma \leq a$ for each element a in subset A. So $\gamma$ is a lower bound for A and is clearly an infimum since it is an upper bound for the set of lower bounds.