Limit superior: proof of properties.

In summary, the conversation discusses the equivalence of two statements involving a sequence {a_n} in R. It is stated that if statement (b) holds, then for every b > a, there exists a value m where a_n < b for all n ≥ m. This leads to the conclusion that limsup a_n ≥ b for every b > a, even though b may be larger than the largest member of the sequence. The explanation for this is that the limsup is a limit of a subsequence, not necessarily the sequence itself.
  • #1
Monochr
3
0
I am only stuck on one part of one fairly long proof.

Homework Statement



Let {a_n} be a sequence in R. Then the following are equivalent:
(a) limsup a_n = a.
(b) For every b > a, a_n < b for all but finitely many n and for every c < a, a_n > c for infinitely many n.

assume that (b) holds. Then for every b > a, there exists m such that an < b for all n ≥ m. Hence sup_n≥m a_n ≥ b. This implies that limsup a_n ≥ b for every b > a so that limsup a_n ≥ a.

The Attempt at a Solution



How does the limsup a_n ≥ b inequality hold? As far as I understand limsup it is a the limit of a monotonically decreasing sequence, sup_n≥m a_n, which cannot have a limit that is larger than the largest member of the sequence. But we know that b is larger. So shouldn't the limsup a_n ≥ b be a strict inequality because of this?
 
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  • #2


Hello,

Thank you for sharing your question on the forum. I understand your confusion regarding the inequality limsup a_n ≥ b. However, in this case, limsup a_n is defined as the supremum of all subsequential limits of the sequence {a_n}. This means that limsup a_n can take on values that are equal to or greater than b, as long as there exists a subsequence of {a_n} that converges to b. In other words, the limsup is not necessarily a limit of the sequence itself, but rather a limit of a subsequence. Therefore, it is possible for limsup a_n to be equal to or greater than b, even if b is larger than the largest member of the sequence.

I hope this clarifies your confusion. If you have any further questions, please let me know. Keep up the good work on your proof and don't hesitate to reach out for help when needed. Science is all about collaboration and learning from each other. Best of luck!
 

What is the definition of limit superior?

The limit superior of a sequence is the smallest number that is greater than or equal to all of the terms in the sequence, regardless of how the terms are arranged.

How is limit superior related to convergence?

If the limit superior of a sequence is equal to the limit of the sequence, then the sequence converges. However, if the limit superior is infinite or does not exist, then the sequence diverges.

Can the limit superior of a sequence be negative?

Yes, the limit superior can be negative if the sequence contains both positive and negative terms. This means that the limit superior can be any number between negative infinity and positive infinity.

What is the difference between limit superior and supremum?

The limit superior of a sequence is the smallest number that is greater than or equal to all of the terms in the sequence, while the supremum is the smallest number that is greater than or equal to all terms in a set. In other words, the supremum is used for sets, while the limit superior is used for sequences.

How is limit superior used in proofs of mathematical properties?

Limit superior is often used in proofs to show that a sequence has a certain property, such as boundedness or convergence. It can also be used to find the limit of a sequence by taking the limit superior of a subsequence. Additionally, limit superior can help determine the behavior of a sequence as it approaches infinity.

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