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