Prove that $\left\{\frac{x_{n}-y_{n}}{x_{n}+y_{n}}\right\}_{n\in\Bbb N}$ converges, given that $\{x_{n}\}_{n\in\Bbb N}$, $\{y_{n}\}_{n\in\Bbb N}$ are sequences that are increasing in $\Bbb R^+$ so that $\{x_{n}-y_{n}\}_{n\in\Bbb N}$ is bounded.
I apologize in advance because I'm an absolute beginner in this kind of website. If you can do this, I would be eternally grateful.
If $\lim_{n\to\infty}x_n=+\infty$ then because of boundedness $\{x_n-y_n\}$ the limit is equal to 0.
Let both sequences are bounded. Because they are increasing, they are convergent to, say, $x$ and $y$ respectively. Hence $x_n+y_n$ tends to $x+y$ and $x_n-y_n$ to $x-y$, which ends the proof.