Let $(x_n), (y_n)$ be two real sequneces such that $z_n:=x_n-y_n\to l\in\mathbb{R}$. Prove that if $(x_n)$ is bounded, then $(x_n)$ and $(y_n)$ are convergent.
I found this assignment in a sample exam but I think the above statement is not correct, because we can take $x_n=y_n=(-1)^n$ which is bounded by $1$ and then $z_n=0$ obviously converge. But $(x_n)$ and $(y_n)$ don't converge. Is there any similar statement that holds? Maybe just a simple assumption missing like $z_n\neq 0$? And how to prove it?