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Limit Superior

  • Definition: Let {a_n} be a sequence of real numbers. Suppose there is a real number U satisfying the following two conditions: i) For every epsilon > 0 there exists an integer N such that n > N implies a_n < U + epsilon. ii) Given epsilon > 0 and given m > 0, there exists an integer n > m such that a_n > U - epsilon. Then U is called the limit superior (or upper limit) of {a_n}.

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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