Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

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In summary, the notation "lim sup" stands for the limit supremum, which is the largest limit point of a sequence. The proof that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n)) is significant because it simplifies the process of finding the limit supremum. To prove this statement, the limit supremum of the sequence must be an upper bound and a limit point of the set {lim sup(y_n), lim sup(z_n)}. This can be generalized to any finite number of sequences, as long as they are bounded above and either increasing or eventually increasing.
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drawar
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Homework Statement


Let [itex](x_{n})[/itex] be a bounded sequence. For each [itex]n \in \mathbb{N}[/itex], let [itex]y_{n}=x_{2n}[/itex] and [itex]z_{n}=x_{2n-1}[/itex]. Prove that
[itex]\lim \sup {x_n} = \max (\lim \sup {y_n},\lim \sup {z_n})[/itex]


Homework Equations





The Attempt at a Solution



Don't know if I'm at the right path but I've tried letting [itex]M= \lim \sup {x_n}[/itex], [itex]M_{1}= \lim \sup {y_n}[/itex], and [itex]M_{2} = \lim \sup {z_n}[/itex] and see that [itex]M \geq \max (M_{1}, M_{2})[/itex]. How do I proceed from here to prove that [itex]M = \max (M_{1}, M_{2})[/itex]? Thank you!
 
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  • #2
Does this observation help?
$\sup {x_n} = \max (\sup {y_n},\sup {z_n})$
 

FAQ: Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

1. What does the notation "lim sup" mean?

The notation "lim sup" stands for the limit supremum, which is the largest limit point of a sequence. It is also known as the upper limit or the least upper bound of a sequence.

2. What is the significance of proving that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))?

This proof is significant because it shows that the limit supremum of a sequence can be determined by taking the maximum of the limit supremums of two other sequences. This can simplify the process of finding the limit supremum of a sequence.

3. How do you prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))?

The proof involves showing that the limit supremum of the sequence x_n is both an upper bound and a limit point of the set {lim sup(y_n), lim sup(z_n)}. This can be done by showing that x_n is eventually larger than both y_n and z_n, and that it is also larger than any other upper bound of the set.

4. Can this statement be generalized to more than two sequences?

Yes, this statement can be generalized to any finite number of sequences. The proof follows the same logic, where the limit supremum of the sequence in question is compared to the limit supremums of all other sequences in the set.

5. Are there any specific conditions or assumptions that need to be met for this statement to hold?

Yes, for this statement to hold, the sequences x_n, y_n, and z_n must all be bounded above. Additionally, the sequences must be either increasing or eventually increasing, meaning that there is some point after which all terms in the sequence are larger than the previous term.

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