Zero-capital nonnegative wealth under a local deflator ($V_0=0,V\ge 0⟹V\equiv 0$)

For a self-financing portfolio whose product with a strictly positive local deflator is a local martingale, nonnegative wealth makes that product a supermartingale. Starting from zero forces it to vanish almost surely at each time. Continuous paths then give simultaneous vanishing at every time on one probability-one event.