All these definitions are equivalent

Which is Appropriate ? it depends on the problem to solve

I think that it is better to introduce them as their names say : For example, the limit superior $\limsup x_n = \lim_m \sup_{n \ge m}\{ x_n, n \ge m\}$. One first determines the sequence of suprema $y_m := \sup\{ x_n , n \ge m\}$, then the limit of the $y_m$, which turns out to be its infimum, since the sequence is non increasing, is the limit superior.

I still believe that one can better introduce the idea of via the more general approach of "Moore-Smith limit" sometimes referred to as "generalized limit".

Here it is:

Consider a (bounded, for a moment) sequence (a_n). Consider the set of its values A:={a_n; n \in N} and the set A' of all accumulation points of A (i.e., the set of all points x in R which are limits of sequences of elements in A, but none of the elements are equal to x). We have:

- liminf (a_n):= inf(A'),

- limsup (a_n):= sup(A').

Amen

I need to correct my previous message:

- liminf (a_n):= inf{all subsequential limits of (a_n)}

- limsup (a_n):= sup{all subsequential limits of (a_n)}

The difference between the above two includes the case of sequences (a_n) being essentially constant.

A remark. There is a notion of limsup and liminf for a sequence of sets A_n defined by:

limsup A_n := commonpart_n ( sum_{m>n} A_m)

liminf A_n := sum_n ( commonpart_{m>n} A_m)

For arbitrary sequence of reals a_n, introducing subsets A_n := (-\infty, a_n ) we can state:

sup[ limsup A_n ] = limsup a_n, . . . . . sup[ liminf A_n ] = liminf a_n.

(it is assumed that supofemptyset := -\infty)

Every way has its advantages and disadvantages.

I prefer the first way since it is the one that can be generalized to define liminf and limsup of functions more easily.

Please assist me here;

Let {#n} and {$n} be sequence of real numbers such that lim {#n}=# and

x~infinite

lim{¥n}=¥

x~infinite.

Show that

a)lim{#n+¥n}=#+¥

b)lim#n¥n=#¥

To Benson Mutisya: Why are you complicating so obvious things using complicated symbols, which in mathematics asually have different meanings?

My phone doesn't have some mathematics symbols thus I had to use alternative symbols.

To Benson Mutisya

can you, please explain in words, what is the meaning of "x~infinite"?

I think that it is worth adding that, contrary to the ordinary sequential – and, by extension, functional - limit, the limit superior is not linear.

Indeed, if the limit superior of (xn) exists in R and additionally, (xn) is unbounded below, then (–xn) is unbouned above; thus it has a subsequence diverging to positive infinity, and then the limit superior of (–xn) is positive infinity; thus the limit superior of (–xn) is not equal to the opposite of the limit superior of (xn), which proves that the limit superior is not linear.

Obviously, the same holds for any sequence (bxn), where b is any negative number.

This non-linear behavior is easily understood if we think of the limit superior as the supremum of the limit set of the examined sequence, since the supremum of the product of a negative number and a non-empty, bounded above set is not equal to the product of the negative number and the supermum of the said set.

Actually, limitsup(x_n) = - limitinf(- x_n). Therefore limitsup is homogenous only for converging sequences.

I agree with Giovanni Dore that each presentation has merits. Particularly, I like the notion of tails because of the nice monotonicity that supremum and infimum have with subsets. The notion involving subsequential limits is also nice, because it defines the limit superior and limit inferior as a supremum and infimum, which are straightforward concepts.

The concepts of limsup (liminf) that we define for numerical sequences, were generalized by the Polish topologist, Kazimierz Kuratowski in the 1920-ies. Amazing, how topology, set theory, and analysis intertwine.

Dear Samuel A. Hokamp,

a presentation of the two limits combining naturally and smoothly - I believe - the two different notions you mention can be found at

https://q4quantum.files.wordpress.com/2021/08/eventboundsliminfsup.pdf

Any feedback would be highly appreciated.

Regards

Spiros Konstantogiannis,

The presentation brings together the two concepts nicely, but I feel the article could benefit from some succinctness.

In particular, it can be shown that the set of subsequential limits S of a bounded real sequence is the intersection of the closures of sets E_k, where each E_k is the range in R of the kth tail. This immediately brings the two ideas together, and your auxiliary lemma follows from the consequent closedness of S. The remaining results fall quickly as well.

Overall, the presentation was clear. Thank you for sharing the article.

Samuel A. Hokamp ,

thank you very much for your suggestions.