Prove that for every sequence {ak} there is no subsequence which converges to a

Question

Prove that for every sequence {ak} there is no subsequence which converges to a number M > lim sup (ak)

Explanation / Answer

let M - lim sup (a_k) = k > 0 Now suppose there is a subsequence which converges to M. let it be a_k_n --> M then choose an epsilon > 0 , s.t. it is less than k. then there exists p > 0 such that ||M - a_k_n|| p but this means that these a_k_n 's lies in the interval (M - epsilon , M + epsilon) which means lim sup (a_n) p which is a contradiction to the fact that : lim sup (a_n) > (a_n) for all n > some m. but here we can choose a_k_n beyond max { p , m }. which is a contradiction to the fact that M > lim sup a_n

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