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Examples 3.4.2(c):

What is inf, sup, lim inf and lim sup for
This sequence is {-1, 2, -3, 4, -5, 6, -7, ...}. You can quickly check, by looking at the definition of lim inf and lim sup and working out the numbers Aj and Bj that: Note that while lim sup and lim inf are not real numbers, they are uniquely defined as plus or minus infinity. The limit of the original sequence, on the other hand, does not exist at all.

Hence, there is a difference between a limit not existing, and a limit that approaches infinity. In the latter sense, lim inf and lim sup will always exist, which is their most useful property.

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