TITLE: If the sum and the product of two sequences converges to zero, does that mean that each sequence converges to zero?
QUESTION [10 upvotes]: If the sum and the product of two sequences converges to zero, 
does that mean that each sequence converges to zero ?
Thanks

REPLY [2 votes]: Let $a_n+b_n=:s_n$, $\ a_n b_n=:p_n$. Then $a_n$, $b_n$ are the two solutions of  the quadratic equation $x^2-s_n x+p_n=0$. It follows that
$$\max\bigl\{|a_n|,|b_n|\bigr\}={1\over2}\left(|s_n|+\sqrt{s_n^2-4p_n}\right)\to0\qquad(n\to\infty)\ .$$