A
Once you get , you are done proving that is a local martingale, whatever the right-continuous adapted process is. You don't have to specify that is a martingale or some kind of product of martingales.
As for proving that are true martingales, you have the following result: if is a local martingale with , the following is equivalent :
(i) is a true square-integrable martingale ( )
(ii) for all , .
Remember that so that you can easily check this criterion in your case.