Upper and Lower Linits (lim sup and lim inf) - Denlinger, Theorem2.9.6 (b) ....

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In summary, the conversation discusses the proof of Theorem 2.9.6 (b) in Charles G. Denlinger's book "Elements of Real Analysis" and how it follows that the limit of a newly defined sequence $\left\{\overline{x_{n}}\right\}_{n=1}^{\infty}$ cannot exceed the value of $B$. The conversation also includes a more concrete proof for this argument.
  • #1
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I am reading Charles G. Denlinger's book: "Elements of Real Analysis".

I am focused on Chapter 2: Sequences ... ...

I need help with the proof of Theorem 2.9.6 (b)Theorem 2.9.6 reads as follows:View attachment 9245
View attachment 9246
In the above proof of part (b) we read the following:

" ... ... Then \(\displaystyle B\) is an upper bound for every \(\displaystyle n\)-tail of \(\displaystyle \{ x_n \}\), so \(\displaystyle \overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B\). Thus \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... ... "My question is as follows:

Can someone please explain exactly how it follows that \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... that is, how it follows that \(\displaystyle \overline{ \lim_{ n \to \infty } } x_n \leq B\) ...
(... ... apologies to steep if this is very similar to what has been discussed recently ... )
Hope someone can help ...

Peter
===============================================================================
It may help MHB readers to have access to Denlinger's definitions and notation regarding upper and lower limits ... so I am providing access to the same ... as follows:
View attachment 9247
View attachment 9248Hope that helps ...

Peter
 

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  • #2
Hi Peter,

Peter said:
Can someone please explain exactly how it follows that \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... that is, how it follows that \(\displaystyle \overline{ \lim_{ n \to \infty } } x_n \leq B\) ...

The newly defined sequence $\left\{\overline{x_{n}}\right\}_{n=1}^{\infty}$ satisfies $\overline{x_{n}}\leq B$ for all $n$. Hence the limit of this sequence cannot exceed the value of $B$. Does this answer your question?
 
  • #3
GJA said:
Hi Peter,
The newly defined sequence $\left\{\overline{x_{n}}\right\}_{n=1}^{\infty}$ satisfies $\overline{x_{n}}\leq B$ for all $n$. Hence the limit of this sequence cannot exceed the value of $B$. Does this answer your question?

Thanks for the help GJA ...

... your argument gives a plausible account of why \(\displaystyle \overline{ x_n } = \text{sup} \{ x_k \ : \ k \geq n \} \leq B\) implies that \(\displaystyle \lim_{ n \to \infty } \overline{ x_n } \leq B\) ... ...

But does your argument constitute an explicit, formal and rigorous argument/proof that a skeptic would accept ...

What do you think ... am I being too extreme or particular ...

Can you help/comment further ...?

Peter
 
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  • #4
A more concrete "proof" would go something like this:

Suppose the limit of the sequence is $L$ and that $L>B$. Let $\varepsilon =\dfrac{L-B}{2}$ and choose $N$ such that $|\overline{x_{n}}-L|<\varepsilon$ for all $n\geq N$. Then $\overline{x_{n}}>L-\varepsilon=\dfrac{L+B}{2}>B$ for all $n\geq N$. This, however, contradicts $\overline{x_{n}}\leq B$ for all $n$.
 
  • #5
GJA said:
A more concrete "proof" would go something like this:

Suppose the limit of the sequence is $L$ and that $L>B$. Let $\varepsilon =\dfrac{L-B}{2}$ and choose $N$ such that $|\overline{x_{n}}-L|<\varepsilon$ for all $n\geq N$. Then $\overline{x_{n}}>L-\varepsilon=\dfrac{L+B}{2}>B$ for all $n\geq N$. This, however, contradicts $\overline{x_{n}}\leq B$ for all $n$.

Thanks for all your help on this issue GJA ...

Peter
 

1. What are upper and lower limits (lim sup and lim inf)?

Upper and lower limits, also known as lim sup and lim inf, are mathematical concepts used to determine the behavior of a sequence or series. They represent the highest and lowest possible values that a sequence or series can approach, respectively.

2. How are upper and lower limits calculated?

Upper and lower limits are calculated by taking the limit of the supremum (lim sup) and infimum (lim inf) of a sequence or series. This means finding the highest and lowest possible values that the sequence or series can approach as the number of terms increases.

3. What is the significance of upper and lower limits in mathematics?

Upper and lower limits are important in mathematics because they help us understand the behavior of a sequence or series. They can also be used to prove the convergence or divergence of a sequence or series, and to determine the exact value of a limit in some cases.

4. How do upper and lower limits relate to Denlinger's Theorem 2.9.6 (b)?

In Denlinger's Theorem 2.9.6 (b), upper and lower limits are used to prove the convergence of a sequence or series. The theorem states that if the lim sup and lim inf of a sequence or series are equal, then the sequence or series converges.

5. Can upper and lower limits be used in real-world applications?

Yes, upper and lower limits have various applications in real-world situations, such as in finance, physics, and engineering. For example, they can be used to analyze the behavior of stock prices, the movement of particles, and the stability of structures.

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