Show existence of subsequences

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In summary, evinda is discussing how to find a subsequence that converges to a given supremum. He suggests modifying the sequence so that the supremum is attained.
  • #1
evinda
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Hello! (Wave)

Let $(a_n)$ be a bounded sequence such that $\inf_{\ell} a_{\ell}<a_n< \sup_{\ell} a_{\ell}$ for each $n=1,2, \dots$

I want to show that there are subsequences $(a_{k_n})$ increasing and $(a_{m_n})$ decreasing such that $a_{k_n} \to \sup_{\ell} a_{\ell}$ and $a_{m_n} \to \inf_{\ell} a_{\ell}$.Could you give me a hint how we could find the desired subsequences? (Thinking)
 
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  • #2
Hey evinda!

I don't think it's true. (Worried)

Suppose we take the sequence $(a_n) = (0,2,1,1,1,1,...)$.
Then it's bounded and its supremum is $2$, but there is no subsequence that converge to $2$ is there? (Wondering)
 
  • #3
Klaas van Aarsen said:
Hey evinda!

I don't think it's true. (Worried)

Suppose we take the sequence $(a_n) = (0,2,1,1,1,1,...)$.
Then it's bounded and its supremum is $2$, but there is no subsequence that converge to $2$ is there? (Wondering)
That sequence attains its supremum (and infimum). evinda is looking at bounded sequences that do not attain their least upper or greatest lower bounds.

@evinda:
This is my suggestion. Let $s=\sup_la_l$. Try and construct a subsequence $\left(a_{k_n}\right)$ as follows. Let $a_{k_1}=a_1$. Then there exists $a_{n_1}$ such that $a_{k_1}<\dfrac{a_{k_1}+s}2<a_{n_1}<s$ since $s$ is the least upper bound. Let $a_{k_2}=a_{n_1}$. Now $\exists a_{n_2}$ such that $a_{k_2}<\dfrac{a_{k_2}+s}2<a_{n_2}<s$; let $a_{k_3}=a_{n_2}$. Continue this way to get the subsequence $\left(a_{k_n}\right)$ where at the $i$th stage $\exists a_{n_i}$ such that $a_{k_i}<\dfrac{a_{k_i}+s}2<a_{n_i}<s$ and you let $a_{k_{i+1}}=a_{n_i}$. Clearly $\left(a_{k_n}\right)$ is increasing and $\dfrac{a_{k_1}+\left(2^n-1\right)s}{2^n}<a_{k_{n+1}}<s$ (easily proved by induction); since $\dfrac{a_{k_1}+\left(2^n-1\right)s}{2^n}\to s$ as $n\to\infty$, $\displaystyle\lim_{n\to\infty}a_{k_n}=s$ by the squeeze theorem.

Similarly for the other subsequence $\left(a_{m_n}\right)$.
 
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  • #4
Hi Everyone,

Wanted to politely jump into say that the construction above needs a slight modification since it does not ensure that the subsequence we select converges to the supremum. Ignoring the infimum stuff, consider the sequence defined by: $a_{2n+1}=-\dfrac{1}{2n+1}$ and $a_{2n}=1-\dfrac{1}{2n}.$ Then it is possible to have selected the subsequence $-1, -1/3, -1/5, \ldots,$ which does not converge to the supremum value of 1.

Note: I admit this example does not possesses the strict inequality property on the infimum, but that is independent of what we are considering here.
 
  • #5
GJA said:
Hi Everyone,

Wanted to politely jump into say that the construction above needs a slight modification since it does not ensure that the subsequence we select converges to the supremum. Ignoring the infimum stuff, consider the sequence defined by: $a_{2n+1}=-\dfrac{1}{2n+1}$ and $a_{2n}=1-\dfrac{1}{2n}.$ Then it is possible to have selected the subsequence $-1, -1/3, -1/5, \ldots,$ which does not converge to the supremum value of 1.

Note: I admit this example does not possesses the strict inequality property on the infimum, but that is independent of what we are considering here.
Thanks GJA. I also noticed the same thing and was editing my post as you posted; I think the edited version will work now. (Nod)
 

Related to Show existence of subsequences

1. What is the significance of showing the existence of subsequences in scientific research?

Showing the existence of subsequences is important because it allows us to identify and study patterns within a larger sequence of data. This can provide valuable insights into the underlying mechanisms and processes at work.

2. How do you determine the presence of a subsequence within a larger sequence?

To determine the presence of a subsequence, we must first identify the pattern or characteristics of the subsequence we are looking for. Then, we can compare the larger sequence to this pattern and see if it matches. This can be done using various mathematical and statistical techniques.

3. Can the existence of subsequences be proven definitively?

No, the existence of subsequences cannot be definitively proven. It is always possible that there are patterns or relationships within a sequence that we have not yet discovered or considered. However, through rigorous analysis and testing, we can provide strong evidence for the existence of certain subsequences.

4. What are some potential applications of studying subsequences in scientific research?

The study of subsequences has many potential applications in scientific research. For example, it can be used in genetics to analyze DNA sequences and identify genetic variations. It can also be used in climate research to identify patterns in temperature or precipitation data. Additionally, it can be applied in economic and financial analysis to identify trends and patterns in stock market data.

5. Are there any limitations to using subsequences in scientific research?

While the study of subsequences can provide valuable insights, there are some limitations to consider. One limitation is that the existence of a subsequence does not necessarily imply causation. Additionally, the results may be influenced by the chosen parameters and methods used to identify the subsequence. It is important to carefully consider these limitations and validate the results through multiple approaches.

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