Lim Sup and Lim Inf

When dealing with sequences there are two choices:

the sequence converges
the sequence diverges

While we know how to deal with convergent sequences, we don't know much about divergent sequences. One possibility is to try and extract a convergent subsequence, as described in the last section. In particular, Bolzano-Weierstrass' theorem can be useful in case the original sequence was bounded. However, we often would like to discuss the limit of a sequence without having to spend much time on investigating convergence, or thinking about which subsequence to extract. Therefore, we need to broaden our concept of limits to allow for the possibility of divergent sequences.

Definition: Lim Sup and Lim Inf

Let be a sequence of real numbers. Define
- A_j = inf{a_j , a_{j + 1} , a_{j + 2} , ...}
and let c = lim (A_j). Then c is called the limit inferior of the sequence .
Let be a sequence of real numbers. Define
- B_j = sup{a_j , a_{j + 1} , a_{j + 2} , ...}
and let c = lim (B_j). Then c is called the limit superior of the sequence .
In short, we have:
1. lim inf(a_j) = lim(A_j) , where A_j = inf{a_j , a_{j + 1} , a_{j + 2} , ...}
2. lim sup(a_j) = lim(B_j) , where B_j = sup{a_j , a_{j + 1} , a_{j + 2} , ...}

When trying to find lim sup and lim inf for a given sequence, it is best to find the first few A_j's or B_j's, respectively, and then to determine the limit of those. If you try to guess the answer quickly, you might get confused between an ordinary supremum and the lim sup, or the regular infimum and the lim inf.

Examples:

What is inf, sup, lim inf and lim sup for ?
What is inf, sup, lim inf and lim sup for ?
What is inf, sup, lim inf and lim sup for

While these limits are often somewhat counter-intuitive, they have one very useful property:

Proposition: Lim inf and Lim sup exist

lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers.

It is important to try to develop a more intuitive understanding about lim sup and lim inf. The next results will attempt to make these concepts somewhat more clear.

Proposition: Characterizing lim sup and lim inf

Let be an arbitrary sequence and let c = lim sup(a_j) and d = lim inf(a_j). Then
1. there is a subsequence converging to c
2. there is a subsequence converging to d
3. d lim inf lim sup c for any subsequence {}
If c and d are both finite, then: given any > 0, there are arbitrary large j such that a_j > c - and arbitrary large k such that a_k < d +

A little bit more colloquial, we could say:

A_j picks out the greatest lower bound for the truncated sequences {a_j}. Therefore A_j tends to the smallest possible limit of any convergent subsequence.
Similarly, B_j picks the smallest upper bound of the truncated sequences, and hence tends to the greatest possible limit of any convergent subsequence.

Compare this with a similar statement about supremum and infimum.

Example

If is the sequence of all rational numbers in the interval [0, 1], enumerated in any way, find the lim sup and lim inf of that sequence.

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