0

If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$

In this question, they have proved that there exists a subsequence $ka_k \to 0$. But we should prove the sequence is Cauchy right? Is Cauchy condensation test really helpful? How to proceed further?

Madhan Kumar
  • 826
  • Proving a sequence is Cauchy only proves it converges, but not what it converges to. The condensation test is about sums, and I suspect it is not helpful. What happens if the sequence $ka_k$ does not converge to zero? Its elements will always be greater than some non-zero $x$. Then you can perhaps compare $a_n$ to $x/n$ and prove by contradiction. – Peter Jul 25 '21 at 11:19
  • The sequence converges to where the subsequence converges. – Madhan Kumar Jul 25 '21 at 11:28
  • If a sequence is convergent than it and every subsequence converge to the same value. That is not the Cauchy condensation test. We know that $\sum\frac{1}{n}$ is not convergent. What do you get if you do the comparison I suggested? – Peter Jul 25 '21 at 11:46

0 Answers0