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Proposition 3.4.6: Lim sup, lim inf, and limit

If a sequence {aj} converges then
lim sup aj = lim inf aj = lim aj
Conversely, 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. Hence
c = lim sup aj = lim aj
To 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, ...) = Aj
Noting 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.

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