Why is the Series Absolutely Convergent for Sequences in l^2(N)?

In summary, the conversation is about a specific example from Houshang H. Sohrab's book "Basic Real Analysis" (Second Edition), focusing on Chapter 2: Sequences and Series of Real Numbers. The example discusses the absolute convergence of a series given two sequences. The question is asking for clarification on how/why the series is absolutely convergent. This is proven using an inequality, , which shows that the series is less than or equal to the product of the two sequences' norms, which is finite.
  • #1
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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 an aspect of Example 2.3.52 ...

The start of Example 2.3.52 reads as follows ... ...
View attachment 9109
In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences , the series is absolutely convergent ... ..."
My question is as follows:

How/why, exactly, given any sequences ...

... does it follow that the series is absolutely convergent ... ...?

Help will be much appreciated ... ...

Peter
 

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  • #2
Peter said:
In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences , the series is absolutely convergent ... ..."
My question is as follows:

How/why, exactly, given any sequences ...

... does it follow that the series is absolutely convergent ... ...?
This comes from the inequality , where it is proved that
 
  • #3
Opalg said:
This comes from the inequality , where it is proved that
Appreciate the help, Opalg ...

Thanks ...

Peter
 

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