lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers.Context
Proposition 3.4.3: Lim inf and Lim sup exist |
lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers. |
Aj = inf{aj , aj + 1 , aj + 2 , ...}is monotone increasing (which you should prove yourself). Hence, lim inf exists (possibly positive infinity).
The sequence
Bj = sup{aj , aj + 1 , aj + 2 , ...}is monotone decreasing (which you should prove yourself). Hence, lim sup exists (possibly negative infinity).
Here we have to allow for a limit to be positive or negative infinity, which is different from saying that a limit does not exist.
