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$a_n, b_n$ - sequences
Suppose $a_n+b_n$ converges. Does $a_n b_n$ converge also?

I tried thinking if I can learn something about $a_n$ and $b_n$ by the assumption $a_n b_n$ converges.
I also tried to develop this equation $|a_n b_n - L| < \epsilon$ assuming it is converging.

I didn't get any bright conclusions.
Will be glad help.

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  • $\begingroup$ What does the first "sentence" mean? Certainly those are sequences, not sets. $\endgroup$
    – Marc van Leeuwen
    Nov 25, 2013 at 13:51

2 Answers 2

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$a_n =n, b_n =-n$

$a_n+b_n=0$ but then...

$a_nb_n=-n^2\rightarrow -\infty$

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    $\begingroup$ When we are talking about convergence, I think it needs to converges to a real number. A sequences that tends to infinity does not converges $\endgroup$
    – Giiovanna
    Nov 25, 2013 at 13:34
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    $\begingroup$ Yes, indeed. This only shows an example when the product does not converges. But it is really easy to show one that the product does converges. Just take 2 constant sequences. The sum is a constant and so does the product. Then, as we can see, the product can or not converge $\endgroup$
    – Giiovanna
    Nov 25, 2013 at 13:39
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    $\begingroup$ Could you please let me know what are you expecting when you say "general way" $\endgroup$
    – user87543
    Nov 25, 2013 at 14:04
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    $\begingroup$ @Giiovanna: The question is if $a_n+b_n$, does $a_nb_n$ also converge? Certainly, anyone can come up with cases where both converge. However, to negate a false implication, one needs to satisfy the hypotheses (in this case, $a_n+b_n$ converges), yet show that the conclusion is not necessarily true (in this case, $a_nb_n$ does not converge). Finding a pair of sequences for which $a_nb_n$ converges does not apply to the question. $\endgroup$
    – robjohn ♦
    Nov 25, 2013 at 15:34
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    $\begingroup$ @Giiovanna If you were asked to prove that 2x is not equal to x^2 how would you do that? Then what would you say to the person that says well it's true for x=2! $\endgroup$
    – Cruncher
    Nov 25, 2013 at 18:01
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As shown in Praphulla Koushik's answer $$ a_n=n, b_n=-n $$ cancellation is a problem.

However, even if you restrict $a_n,b_n\ge0$ then the answer is no. For example, if $$ a_n=2+(-1)^n,b_n=3-(-1)^n $$ then $a_n+b_n=5$ yet $a_nb_n$ oscillates between $4$ and $6$.

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