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Examples 3.4.2(a):

What is inf, sup, lim inf and lim sup for ?
Clearly, the infimum of the sequence is -1, and the supremum is +1. To find lim inf and lim sup, we will first find the sequence of numbers Aj and Bj mentioned in the definition.

Let's find the numbers Aj = inf{aj, aj + 1, aj + 2, ...} for the sequence {-1, 1, -1, 1, ...}.

and so on. Therefore, it is clear that
lim inf = -1
Similarly, we find the numbers Bj = sup{aj, aj + 1, Aj + 2, ...}: and so on. Therefore, it is clear that
lim sup= 1
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