## Proposition 3.4.4: Characterizing lim sup and lim inf

*c = lim sup a*and

_{j}*d = lim inf a*. Then

_{j}- there is a subsequence converging to
*c* - there is a subsequence converging to
*d* -
*d lim inf lim sup c*for any subsequence*{}*

*c*and

*d*are both finite, then: given any

*> 0*there are arbitrary large

*j*such that

*a*and arbitrary large

_{j}> c -*k*such that

*a*.

_{k}< d +

### Proof:

First let's assume that*c = lim sup{a*is finite, which implies that the sequence

_{j}}*{a*is bounded. Recall the properties of the sup (and inf) for sequences:

_{j}}If a sequence is bounded above, then given anyNow take any> 0there exists at least one integerksuch thata_{k}> c -

*> 0*. Then

so by the above property there exists an integerA_{k}= sup{a_{k}, a_{k+1}, ...}

*j*such that

_{k}> kor equivalentlyA_{k}> > A_{k}- / 2

We also have by definition that| A_{k}- | < / 2

*A*converges to

_{k}*c*so that there exists an integer

*N*such that

But now the subsequence| A_{k}- c | < / 2

*{}*is the desired one, because:

if| - c | = | - A_{k}+ A_{k}- c |

| - A_{k}| + | A_{k}- c |

< / 2 + / 2 =

*j*. Hence, this particular subsequence of

_{k}> N*{a*converges to

_{n}}*c*.

The proof to find a subsequence converging to the *lim inf* is
similar and is left as an exercise.

Statement (3) is pretty simple to prove: For any sequence we always have that

Taking limits on both sides givesinf{a_{k}, a_{k+1}, ... } sup{a_{k}, a_{k+1}, ... }

*lim inf(a*for any sequence, so it is true in particular for any subsequence.

_{n}) lim sup(a_{n})
Next take any subsequence of *{a _{n}}*. Then:

because an infimum overinf(a_{k}, a_{k+1}, ...) inf(, , ...)

*more*numbers (on the left side) is less than or equal to an infimum over fewer numbers (on the right side). But then

The proof of the inequalityd lim inf()

*lim sup() c*is similar. Taking all pieces together we have shown that

for any subsequenced lim inf lim sup c

*{}*, as we set out to do.

It remains to show that given any
*> 0* there are
arbitrary large *j* such that
*a _{j} > c - *
(as well as the corresponding statement for the

*lim inf d*).

But previously we have found a subsequence
*{}* that
converges to *c* so that there exists an integer
*N* such that

if| - c | <

*k > N*. But that means that

*- < - c <*which implies that

as long asc - < < c +

*k > N*. But that of course means that there

*are*arbitrarily large indices - namely those

*j*for which

_{k}*k > N*- with the property that

*> c -*as required. Hence, we have shown the last statement involving the

*lim sup*, and a similar proof would work for the

*lim inf*.

All our proofs rely on the fact that the *lim sup* and
*lim inf* are bounded. It is not hard to adjust them for
unbounded values, but we will leave the details as an exercise.

Contributed to this page:*
Thomas Wollmann
*