1

Suppose $\{a_i\}$ is a sequence in $\mathbb{R}$ such that $\sum|a_i||x_i| < \infty$ whenever $\sum |x_i| < \infty$. Then $\{a_i\}$ is a bounded sequence.

Does $\sum\limits_{i=1}^{\infty}|a_i||x_i| < \infty$ whenever $\sum\limits_{i=1}^{\infty} |x_i| < \infty $ imply $(a_i)$ is bounded? This was asked before. My question is different. I want to know whether we can proceed by this method/idea?

$\sum|a_i||x_i| < \infty$ implies $|a_i||x_i|\to 0$ and $\sum |x_i| < \infty$ implies $|x_i|\to 0$

Hypothesis " $\mathbb{R}$ such that $\sum|a_i||x_i| < \infty$ whenever $\sum |x_i| < \infty$" can be interpreted as $|a_i||x_i|\to 0$ whenever $|x_i|\to 0$

Can we use this info to given an alternate proof.

Thanks for reading.

Shweta Aggrawal
  • 5,501
  • 2
  • 15
  • 48

1 Answers1

3

Convergence of a series $\sum a_n$ of positive terms cannot be determined by just seeing if $a_n \to 0$. This condition is necessary but not sufficient for convergence. So you cannot give a simpler proof by just looking at the limits of the general terms of the series.

Kavi Rama Murthy
  • 311,013