The **limit inferior and limit superior** of a sequence give bounds on the sequence’s subsequential limits (i.e. the limit of a subsequence) [1]:

- The
**limit inferior**is the*smallest*limit of a subsequence. - The
**limit superior**is the*largest*limit of a subsequence.

For bounded sequences, these limits always exist [2].

In notation, the *limit inferior* may be written as:

The *limit superior* may be written as:

## Visual Example of Limit Inferior and Limit Superior

The bounded sequence a_{i} = (-1)^{i}(i + 1)/i is not convergent.

However, two subsequences (odd terms and even terms) are convergent, as shown on the following graph:

From the graph, we can see that the subsequence of even terms converges to 1, which means that the limit superior = 1. The limit inferior converges from below to -1. These limits give a qualitative measure of a sequence’s asymptotic behavior [3].

Another example is the (relatively) famous divergent sequence (1, -1, 1, -1, 1,…). While the sequence as a whole does not converge, the even terms converge to -1 (i.e. lim inf = 1) and the odd terms converge to 1 (lim sup = 1).

## Formal Definition

The limit inferior for a sequence x_{n}can more formally be defined as follows:

The limit superior can be defined in a similar way:

The limit inferior is always smaller than the limit superior, unless the sequence is convergent. If that happens, then the two limits are equal. In notation, we can say:

lim inf ≤ lim sup.

## References

Sequence Images: Desmos.

[1] Basic properties of limit inferior and limit superior. Retrieved May 3, 2021 from: https://www.uio.no/studier/emner/matnat/math/MAT1100/h20/grublelimsup.pdf

[2] Lebl, J. Basic Analysis I & II: Introduction to Real Analysis, Volumes I & II. 2.3 Limit superior, limit inferior, and Bolzano–Weierstrass. Retrieved May 3, 2021 from: https://www.jirka.org/ra/html/sec_bw.html

[3] Chidume Chapter 5. Retrieved May 3, 2021 from: http://www.math.utoledo.edu/~dwhite1/d_makerere/Chidume2.pdf

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