Let $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ be sequences in $\mathbb{R}$ such that $x_n \to \pm \infty$ and $(x_ny_n)_{n \in \mathbb{N}}$ converges. Show that $y_n \to 0$.
This problem is driving me crazy. All I did so far was to state the following definitions:
If $x_n \to \infty$, then for all $M > 0$, there exists an $n_1 \in \mathbb{N}$ such that $n \ge n_1$ implies $x_n > M$.
If $x_n \to -\infty$, then for all $N < 0$, there exists an $n_2 \in \mathbb{N}$ such that $n \ge n_2$ implies $x_n < N$.
If $x_ny_n \to L$, where $L \in \mathbb{R}$, then for all $\epsilon > 0$, there exists an $n_3 \in \mathbb{N}$ such that $n \ge n_3$ implies $|x_n y_n-L|<\epsilon$.
How can I approach this problem? I thought about proving by contradiction, that is, what happens if $L > 0$ and if $L < 0$, for each case of $x_n \to +\infty$ and $x_n \to -\infty$?