On a dyadic approximation of predictable processes of finite variation

We show that any càdlàg predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated ‘from below’ by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.