PAC implies convergence in probability but is more powerful. PAC results are non-asymptotic (and in supremum) so for a fixed n you can chose a degree of uncerta

PAC implies convergence in probability but is more powerful. PAC results are non-asymptotic (and in supremum) so for a fixed n you can chose a degree of uncertainty and get the corresponding bound. Usual convergence results in statistics give you the asymptotic rate of convergence at best, nothing else. That's why PAC / VC results are so useful for ML: you get results for the useful case (limited and fixed amount of samples, estimator that isn't even in the same class of function as the true function), not the theoretical case where you somehow managed to get infinitely many observations. Of course the price to pay for hypothesis that are so relaxed is weaker results.