I am only stuck on one part of one fairly long proof.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let {a_n} be a sequence in R. Then the following are equivalent:

(a) limsup a_n = a.

(b) For every b > a, a_n < b for all but finitely many n and for every c < a, a_n > c for infinitely many n.

assume that (b) holds. Then for every b > a, there exists m such that an < b for all n ≥ m. Hence sup_n≥m a_n ≥ b. This implies that limsup a_n ≥ b for every b > a so that limsup a_n ≥ a.

3. The attempt at a solution

How does the limsup a_n ≥ b inequality hold? As far as I understand limsup it is a the limit of a monotonically decreasing sequence, sup_n≥m a_n, which cannot have a limit that is larger than the largest member of the sequence. But we know that b is larger. So shouldn't the limsup a_n ≥ b be a strict inequality because of this?

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# Homework Help: Limit superior: proof of properties.

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