Two problems I think. First, the kind of bear and the age of the bear. The size and strength of bears range widely. Fighting a sun bear is somewhat reasonable. Fighting a grizzly, on the other hand.....
Second, video proof. I know this kind of things probably aren't taped, because the people are busy fighting for their life. But, still, I'd be skeptical before I see some proof, that a person fought and killed an healthy adult grizzly (or some other comparable types) without a firearm or trap.