## Examples 3.4.2(c):

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

*A*and

_{j}*B*that:

_{j}*inf { (-1)*^{j}j } = -*lim inf { (-1)*^{j}j } = -*sup{ (-1)*^{j}j } =*lim sup{ (-1)*^{j}j } =

*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.