See 3.12 of Williams.
Let where and
- and .
- is right continuous.
Then by the proof . Thus and is therefore an upper bound (if not then such that and but by monotonicity ). Therefore .
On the other hand suppose then (if not then but then is a lower bound for all such which would imply ). Now is an upper bound for any such that so . Now suppose then and by right continuity this implies . Thus we must have