3.4. Lim Sup and Lim Inf
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 Aj and Bj that:- inf { (-1) j j } = -
- lim inf { (-1) j j } = -
- sup{ (-1) j j } =
- lim sup{ (-1) j j } =
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.