Proposition 2.3.15: Understanding Sohrab's Basic Real Analysis Proof

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In summary, Peter is trying to figure out how the proposition 2.3.15 follows from the previous propositions. He is unsure how to proceed due to the lack of a statement of what is required in order for the proposition to follow.
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with yet another aspect of the proof of Proposition 2.3.15 ...Proposition 2.3.15 and its proof read as follows:
View attachment 9073
At the end of the above proof by Sohrab we read the following:

" ... ... The proposition now follows from \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) and Proposition 2.2.39 (f) ... ... "But ... as far as I can tell from Proposition 2.2.39 (f) we require \(\displaystyle \text{ lim sup } (t_n) = e = \text{ lim inf } (t_n)\) ... in order to conclude \(\displaystyle \text{ lim } (t_n) = e\) ...

... but we only have the condition \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) ... ...

So ... how does the proposition 2..3.15 follow ...?

Can someone please clarify the situation above ...
Help will be appreciated ...

Peter
=========================================================================The above post refers to Proposition 2.2.39 ... so I am providing text of the same ... as follows ... :
View attachment 9074
Hope that helps ...

Peter
 

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  • #2
Peter said:
At the end of the above proof by Sohrab we read the following:

" ... ... The proposition now follows from \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) and Proposition 2.2.39 (f) ... ... "But ... as far as I can tell from Proposition 2.2.39 (f) we require \(\displaystyle \text{ lim sup } (t_n) = e = \text{ lim inf } (t_n)\) ... in order to conclude \(\displaystyle \text{ lim } (t_n) = e\) ...

... but we only have the condition \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) ... ...

So ... how does the proposition 2..3.15 follow ...?
It is always true that $\liminf (t_n) \leqslant \limsup(t_n)$. So in the inequalities $ \limsup(t_n) \leqslant e \leqslant \liminf (t_n) \leqslant \limsup(t_n)$ the last term is the same as the first, and therefore there must be equality throughout.
 
  • #3
Opalg said:
It is always true that $\liminf (t_n) \leqslant \limsup(t_n)$. So in the inequalities $ \limsup(t_n) \leqslant e \leqslant \liminf (t_n) \leqslant \limsup(t_n)$ the last term is the same as the first, and therefore there must be equality throughout.
Thanks for the help, Opalg ...

Peter
 

FAQ: Proposition 2.3.15: Understanding Sohrab's Basic Real Analysis Proof

What is Proposition 2.3.15 in Sohrab's Basic Real Analysis Proof?

Proposition 2.3.15 is a statement in Sohrab's Basic Real Analysis Proof that states: "If a sequence of real numbers converges, then it is bounded." This means that if a sequence of real numbers approaches a specific limit, then the numbers in the sequence must be contained within a certain range or interval.

Why is Proposition 2.3.15 important in Sohrab's Basic Real Analysis Proof?

This proposition is important because it helps to prove the completeness property of the real numbers. It shows that if a sequence of real numbers converges, then it must be bounded, which is a key concept in understanding the behavior of real numbers.

How does Proposition 2.3.15 relate to other concepts in real analysis?

Proposition 2.3.15 is closely related to the concepts of convergence and boundedness in real analysis. It also plays a role in proving other important theorems, such as the Bolzano-Weierstrass theorem.

What is the proof for Proposition 2.3.15?

The proof for Proposition 2.3.15 involves using the definition of convergence and the Archimedean property of the real numbers. It can be broken down into several steps, including showing that the sequence is bounded above and below, and using the definition of convergence to show that the sequence is bounded within a specific range.

How can Proposition 2.3.15 be applied in real-world scenarios?

Proposition 2.3.15 can be applied in various real-world scenarios, such as in economics, physics, and engineering. For example, in economics, it can be used to analyze the behavior of financial markets and stock prices. In physics, it can be applied to understand the motion of particles and the behavior of physical systems. In engineering, it can be used to analyze the stability and efficiency of systems and structures.

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