TITLE: Real sequences which sum to 0, multiply to 1.
QUESTION [11 upvotes]: Does there exist two sequences $(x_n)_n$, $(y_n)_n$ of real numbers such that
$\lim_n x_n-y_n\neq 0$ (may not exist), but $\lim_n x_n+y_n=0$ and $\lim_n x_ny_n=1$?
Notice that it cannot be the case that both sequences are convergent, or even bounded, since by taking convergent subsequences, say with limits $x$ and $y$, we would have that $x+y=0$ and $xy=1$, which has no real solutions.

REPLY [2 votes]: Assume toward a contradiction that there are such sequences $(x_n)_n$ and $(y_n)_n$.  Because $\lim_n x_n y_n$ is positive, $x_n$ and $y_n$ eventually have the same sign.  Without loss of generality, we may assume that for infinitely many $n$ they are both positive.  By passing to a subsequence we may assume that $x_n$ and $y_n$ are both positive for all $n$.  Then $x_n$ and $y_n$ are both between $0$ and $x_n + y_n$, which approaches zero as $n \to \infty$, so $x_n$ and $y_n$ themselves approach zero as $n \to \infty$.  Therefore $\lim_n x_ny_n = 0$, which is a contradiction.