Basically.

You could do a bit better. E.g. if you were confident that you could oversample “attacks” by a factor of X, then you can draw N samples and then be confident about everything with probability > 1/NX.

In any case, yes you would end up with a distribution with finite support (but “finite” has a lot of wiggle room, if iterated amplification works at all there is clearly some finite support that is adequate).