I have the following question in my assignment which I couldn't solve:

Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $

I have to prove if the following is true or false:

$\lim\limits_{n \to \infty}({a_n}) =0$ $or$ $\lim\limits_{n \to \infty}({b_n}) =0$

It seems true to me, for several reasons, but mainly because I couldn't find an example that contradicts the statement. I can not use limit arithmetic because I can't prove that the sequences are convergent, so every time I tried to do some calculation, I got stuck.

Your help is appreciated, thank you.

  • 6
    $\begingroup$ $1,0,1,0,1,\dots$ and $0,1,0,1,0,\dots$. $\endgroup$ – André Nicolas Mar 25 '15 at 6:23
  • $\begingroup$ @AndréNicolas thank you! Sometimes I feel so stupid not solving these questions. $\endgroup$ – Alan Mar 25 '15 at 6:26
  • 2
    $\begingroup$ You are welcome. Another idea added to the bag of tricks! $\endgroup$ – André Nicolas Mar 25 '15 at 6:38

Hint: Consider the sequences $a_n=(-1)^n+1$ and $b_n=(-1)^{n+1}+1$. You should notice that $a_n\cdot b_n=0$ for all $n$, and yet neither series is convergent as $n\rightarrow+\infty$.


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