## Proposition 3.4.6: Lim sup, lim inf, and limit

If a sequence

*{a*converges then_{j}}Conversely, iflim sup a_{j}= lim inf a_{j}= lim a_{j}

*lim sup a*are both finite then_{j}= lim inf a_{j}*{a*converges._{j}}### Proof:

Let*c = lim sup a*. From before we know that there exists a subsequence of

_{j}*{a*that converges to

_{j}}*c*. But since the original sequence converges, every subsequence must converge to the same limit. Hence

To prove thatc = lim sup a_{j}= lim a_{j}

*lim inf a*is similar.

_{j}= cThe converse of this statement can be proved by noting that

Noting thatB_{j}= inf(a_{j}, a_{j+1}, ...) a_{j}sup(a_{j}, a_{j+1}, ...) = A_{j}

*lim B*we can apply the Pinching Theorem to see that the terms in the middle must converge to the same value.

_{j}= lim inf(a_{j}) = lim sup(a_{j}) = lim A_{j}