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Priyadarshan Choursiya Best Answer
if odd and even sub sequence of a sequence converge to same no. then original sequence is convergent here you see that even sub sequence of An is increasing and bounded above by A1, similarly odd sub sequence of An is decreasing and bounded below by A2, thus both the subsequences are convergent. given limit of Bn is zero which imply odd and even sub sequence have same limit. thus the seq An is convergent.