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- Jun 22, 2012

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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 ... ...

In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\), the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ..."

My question is as follows:

How/why, exactly, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\) ...

... does it follow that the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ...?

Help will be much appreciated ... ...

Peter