# The Class of All Square-Summable Real Sequences... l^2(N) ... Sohrab Example 2.3.52 ... ...

#### Peter

##### Well-known member
MHB Site Helper
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 ... ...

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

#### Opalg

##### MHB Oldtimer
Staff member
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 ... ...?
This comes from the inequality $(\dagger)$, where it is proved that $$\displaystyle \sum_{ n = 1 }^{ \infty }| x_n y_n| \leqslant \|x\|_2\|y\|_2 < \infty.$$

#### Peter

##### Well-known member
MHB Site Helper
This comes from the inequality $(\dagger)$, where it is proved that $$\displaystyle \sum_{ n = 1 }^{ \infty }| x_n y_n| \leqslant \|x\|_2\|y\|_2 < \infty.$$

Appreciate the help, Opalg ...

Thanks ...

Peter