Predictability, local boundedness and martingales.

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.

-
What is $K$ here? –  Davide Giraudo Nov 16 '12 at 20:57
the uniform bound for the process. –  Lost1 Nov 18 '12 at 11:05
@DavideGiraudo see post above. only realised how to tag. I'd put bounty on this question until it is answered... if I could... and had the rep to do that. –  Lost1 Nov 26 '12 at 0:22