TITLE: Suppose $a_n+b_n$ converges. Does $a_n*b_n$ converges also?
QUESTION [5 upvotes]: $a_n, b_n$ - 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.

REPLY [15 votes]: 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 $6$.