Lemma 2 proves (1/n^{1/a_n}) sum Y_k -> 0, but the theorem needs (1/n) sum Y_k -> 0, and a counterexample shows the claimed SLLN fails.
Cantelli, Sulla probabilit` a come limite della frequenza, Atti Ac- cad
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On strong law of large numbers for non identically distributed random variables
Lemma 2 proves (1/n^{1/a_n}) sum Y_k -> 0, but the theorem needs (1/n) sum Y_k -> 0, and a counterexample shows the claimed SLLN fails.