How to Prove Convergence in Sequence $(x_n)_{n=1}^\infty$?

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  • Thread starter Euge
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    2016
In summary, POTW #231 is a mathematical problem published by Project Euler on November 1, 2016. The goal of this problem is to find a unique solution that has been verified by the creators of Project Euler. There are guidelines and restrictions set by Project Euler for solving POTW #231, such as not using external resources and providing a self-contained solution. Anyone can submit a solution to POTW #231, but it must meet the guidelines and restrictions to be accepted as valid.
  • #1
Euge
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Here is this week's POTW:

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Consider a sequence of real numbers $(x_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty \lvert x_n y_n\rvert$ converges for every real sequence $(y_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty y_n^2$ converges. Prove that $\sum\limits_{n = 1}^\infty x_n^2$ converges.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
This week's problem was solved correctly by Kokuhaku. Here is the solution.
Let $x = \{x_n\}_{n=1}^\infty$ be fixed and define $A_n(y)=\sum_{k=1}^n x_k y_k$ for $y = \{y_n\}_{n=1}^\infty \in \ell^2(\mathbb{N})$. We have that $A_n$ is pointwise bounded, since $\sup_n |A_n(y)| \leqslant \sum_{n=1}^\infty |x_n y_n| < +\infty$, for every $y \in \ell^2(\mathbb{N})$. Thus, by uniform boundedness principle we have $\sup_n \|A_n\| < +\infty$.

Also, Cauchy-Schwarz inequality (or Hölder inequality with $p,q=2$) gives us $|A_n(y)| \leqslant \sqrt{\sum_{k=1}^n x_k^2} \cdot \|y\|_2$ for $y \in \ell^2(\mathbb{N})$, that is $\|A_n\| \leqslant \sqrt{\sum_{k=1}^n x_k^2}$. Choosing $y_k=x_k$ for $1 \leqslant k \leqslant n$ and $y_k=0$ for $k>n$ we obtain $A_n(y)=\sum_{k=1}^n x_k^2$, and so $\sqrt{\sum_{k=1}^n x_k^2} = \frac{A_n(y)}{\sqrt{\sum_{k=1}^n x_k^2}} \le \|A_n\|$.

So we find that $\|A_n\| = \sqrt{\sum_{k=1}^n x_k^2}$ and then from $\sup_n \|A_n\| < +\infty$ we have that $x \in \ell^2(\mathbb{N})$, since $\|x\|_2 = \sup_n \sqrt{\sum_{k=1}^n x_k^2}$.

Remark: If we have complex sequences, then we would choose $y_k=\operatorname{sign} x_k \cdot |x_k|$ for $k \leqslant n$ and $y_k=0$ for $k>n$. Also, if we have $y \in \ell^p$ and same condition $\sum_{n=1}^\infty |x_n y_n| < +\infty$, then we would had $x \in \ell^q$ (use general Hölder and $y_k=\operatorname{sign} x_k \cdot |x_k|^{q-1}$).
 

Related to How to Prove Convergence in Sequence $(x_n)_{n=1}^\infty$?

1. What is POTW #231 and when was it published?

POTW #231 is a problem from the online problem-solving community Project Euler. It was published on November 1, 2016.

2. What is the goal of POTW #231?

The goal of POTW #231 is to find the solution to a mathematical problem, which is typically a complex mathematical equation or puzzle.

3. Is the solution to POTW #231 a unique answer?

Yes, the solution to POTW #231 is a unique answer that has been verified by the creators of Project Euler.

4. Are there any restrictions or guidelines for solving POTW #231?

Yes, Project Euler has a set of guidelines and restrictions for solving POTW #231. These include not using external resources or programs, and providing a self-contained solution that can be run by others.

5. Can anyone submit a solution to POTW #231?

Yes, anyone can submit a solution to POTW #231 on the Project Euler website. However, only solutions that meet the guidelines and restrictions will be accepted as valid.

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