I have: $$\lim_{n\to \infty}(|x_n+y_n|-|x_n-y_n|)=+\infty$$ Need to prove: $$\lim_{n\to \infty}|x_n|=\lim_{n\to \infty}|y_n|=\lim_{n\to \infty}x_ny_n=+\infty$$ I can prove $\lim_{n\to \infty}|x_n|=\lim_{n\to \infty}|y_n|=+\infty$ but I don't know what to do with other part. I can notice that $\lim_{n\to \infty}|x_n||y_n|=\lim_{n\to \infty}|y_nx_n|=+\infty$ but it's not what I exactly need because $\lim_{n\to \infty}|y_nx_n|=+\infty \Rightarrow \lim_{n\to \infty}y_nx_n=\infty $. Maybe there is some possibility to prove that $ x_n$ and $y_n$ have the same sign.
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For $n$ large enough we must have $|x_n+y_n|-|x_n-y_n|>0$. Suppose $x_n$ and $y_n$ have different signs $sgn$, then we get $$\begin{align}|x_n+y_n|-|x_n-y_n|=&|sgn(x_n)|x_n|-sgn(x_n)|y_n||-|sgn(x_n)|x_n|+sgn(x_n)|y_n||\\=&||x_n|-|y_n||-||x_n|+|y_n||\\\leq& 0\end{align}$$ a contradiction. So for $n$ lage enough we have $x_ny_n>0$, what was to show.