3.4. Lim Sup and Lim Inf
Proposition 3.4.6: Lim sup, lim inf, and limit
If a sequence {aj} converges then
lim sup aj = lim inf aj = lim ajConversely, if lim sup aj = lim inf aj are both finite then {aj} converges.
Proof:
Let c = lim sup aj. From before we know that there exists a subsequence of {aj} that converges to c. But since the original sequence converges, every subsequence must converge to the same limit. Hencec = lim sup aj = lim ajTo prove that lim inf aj = c is similar.
The converse of this statement can be proved by noting that
Bj = inf(aj, aj+1, ...) aj sup(aj, aj+1, ...) = AjNoting that lim Bj = lim inf(aj) = lim sup(aj) = lim Aj we can apply the Pinching Theorem to see that the terms in the middle must converge to the same value.