In George Lowther's Blog he is trying to prove the Debut-Theorem for cadlag-processes.
https://almostsure.wordpress.com/2009/11/15/stopping-times-and-the-debut-theorem/
In the continuous case he is using the Bolzano Weierstraß Theorem in order to justify using the maximum process,(someone asked before):
However I don't understand why the same argumentation cannot be used for Cadlag processes. To be more specific, why should a cadlag function not attain its maximum and minimum in a compact interval? According to the extreme value theorem for semi continuous function, this should be possible, at least for a half of boundedness.