Limits of Sequences .... Bartle and Shebert, Example 3.4.3 (b) ....

In summary, The conversation is about understanding a specific example in "Introduction to Real Analysis" (Fourth Edition) by Robert G. Bartle and Donald R. Sherbert. The example discusses sequences and series and shows that if , then and for all . The conversation includes a request for a rigorous proof of this result and a discussion about the use of definitions in the book. The book aims to develop real analysis rigorously from first principles, but in this case, a well-known definition (1+1=2) is
  • #1
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I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ...

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

I need help in fully understanding Example 3.4.3 (b) ...Example 3.4.3 (b) ... reads as follows:https://www.physicsforums.com/attachments/7230
In the above text from Bartle and Sherbert we read the following:

" ... ... Note that if then and for all . (Why?) ... "Can someone help me to show rigorously that and for all ... ... ?Hope that someone can help ...

Peter
 
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  • #2
Peter said:
In the above text from Bartle and Sherbert we read the following:

" ... ... Note that if then and for all . (Why?) ... "Can someone help me to show rigorously that and for all ... ... ?
If then . Raise both sides to the power to get . In other words,
 
  • #3
Opalg said:
If then . Raise both sides to the power to get . In other words,

Thanks Opalg ... but ... can you indicate how we prove that ... ?

Peter
 
  • #4
Peter said:
Thanks Opalg ... but ... can you indicate how we prove that ... ?

Peter
Suppose that . Then , , ... , and by induction for all In particular, . But that contradicts the fact that . So the initial supposition was wrong, and therefore
 
  • #5
Peter said:
Thanks Opalg ... but ... can you indicate how we prove that ... ?

Peter

 
  • #6
solakis said:
This problem comes from the book by Bartle and Sherbert, which aims to develop real analysis rigorously from first principles. At this stage of the book the logarithm function has not yet been introduced, so it is not available for use.
 
  • #7
Opalg said:
This problem comes from the book by Bartle and Sherbert, which aims to develop real analysis rigorously from first principles. At this stage of the book the logarithm function has not yet been introduced, so it is not available for use.

In another thread the book asks us in exercise 2.1.12(1) to prove :



To prove that we need the definition 1+1=2

Does the book mention that definition anywhere ?

If the answer is yes then i stop the discussion here

But if the answer is no, then one may wonder how rigorously the book aims to develop real analysis.

However one may say ,but 1+1=2 is so well known that the book may omit such obvious definition.

In the same way lun function is so well known even in high school that we may use it,
although the book examines it in more details in later sections
 

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