Notice that for the condition to be satisfied, it is not possible that for each "n" the random variables "X" and are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless "X" is deterministic like for the weak law of large numbers. At the same time, the case of a deterministic "X" cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.