How do you read lim sup and lim inf?

How do you read lim sup and lim inf?

The superior limit is the larger of the two, and the inferior limit is the smaller of the two. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).

How do you find the lim sup of a set?

For a sequence of subsets An of a set X, the lim supAn =∩∞N=1(∪n≥NAn) and lim infAn =∪∞N=1(∩n≥NAn).

Do lim sup and lim inf always 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.

What is sup Xn?

Let {xn} ⊆ R be bounded, then. (1) The limit superior of {xn} is the infimum of V ⊆ R such that xn > v ∈ V for at most a finite number of n ∈ N, and (2) The limit inferior of {xn} is the supremum of W ⊆ R such that xn > w ∈ W for at most a finite number of n ∈ N. These are denoted as lim sup xn and lim inf xn.

Why does lim sup always exist?

The limit of a bounded sequence need not exist, but the liminf and limsup of a bounded sequence always exist as real numbers. When there’s no loss of clarity, we might omit the subscript variable (above, it is n). There are also shorter notations meaning the same thing: liman means lim supan and liman means lim inf a.

Does sup always exist?

Maximum and minimum do not always exist even if the set is bounded, but the sup and the inf do always exist if the set is bounded. If sup and inf are also elements of the set, then they coincide with max and min. Given a set X ⊆ R, if max X exists it is equal to sup X. Proof.

What are lim sup and lim inf of a sequence of sets?

I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets A n of a set X, the lim sup A n = ⋂ N = 1 ∞ ( ⋃ n ≥ N A n) and lim inf A n = ⋃ N = 1 ∞ ( ⋂ n ≥ N A n).

What does Limlim sup A N C mean?

lim sup A n are all the people who don’t get another job. (Categories 2 and 3). Thus, lim inf A n C are all the people who eventually get a new job. (Category 1) lim inf A n are the people who become weekly regulars.

How to prove the equivalence between convergence and equality of lim inf with lim sup?

Proof. From Theorem 1.1 we know that lim infsn= min(S)ax(S) = lim supsn.Now let us prove the equivalence between convergence and equality of lim inf with lim sup. Ifthe sequence is convergent toL, then we know that any subsequence can only converge toL. Itfollows thatS=fLg, hence min(S) = max(S) =L.

What is a lim inf a N C?

Lastly there are the people who have low self-esteem; they feel inferior, and at one point, they don’t care anymore and start to get their food from the church each week. lim sup A n are all the people who don’t get another job. (Categories 2 and 3). Thus, lim inf A n C are all the people who eventually get a new job. (Category 1)