Example:
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:
- inf { (-1) j j } =
-
- lim inf { (-1) j j } =
-
- sup{ (-1) j j } =
- lim sup{ (-1) j j } =
Note that while the 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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