no duplicates then clearly distinct x x this is proved by induction on j when j i item v i was just added to the sample with probability size j so it is in the sample with probability size j by definition if at time j u2212 1v i was in the sample with probability size j u2212 1 then at time j with probability a 1 u2212 size j u2212 1 v i was not in the sample before and is still not in the sample with probability b size j u2212 1 size j 1 size v i was in the sample before and was just evicted otherwise v i is in the sample at time j this has probability 1 u2212 a b 1 u2212 1 u2212 size j u2212 1 size j u2212 1 size j 1 size size j u2212 1 u2212 size j u2212 1 size j 1 size size j u2212 1 u2212 size j u2212 1 j j size u2212 size j j u2212 1 size j u2212 1 j j u2212 1 size j qed 2 duplicates in the v i list have no effect given a v list with duplicates find the first pair i and j such that v i v j and i lt j by removing v j from the list a list v u2032 is constructed that contains one less duplicate pair this shows that the algorithm produces the same results on v and on v u2032 by induction it produces the same.