I would like to make clear on definition of locally bounded. We say a process is locally bounded if there exists a sequence of stopping times such that $X$ stopped at the stopping time is uniformly bounded. This means $X_t(\omega)\leq K$ for almost every $\omega$
I can see why all continuous predictable processes in continuous time must be locally bounded, just by taking the stopping times to be the first time they hit $+ n$ or $-n$.
But is this true for general predictable processes?
Why are all predictable processes in discrete settings locally bounded? I cannot see why the stopping sequence for continuous time and continuous processes still hold.