# Prove convergence and limits for an increasing and a decreasing sequences

Let $(x_n)$, $(y_n)$ be two sequences of real numbers such that $(x_n)$ is an increasing sequence and $(y_n)$ is a decreasing sequence with the property that for all $n \in \mathbb{N}$, $0 \leq y_n - x_n \leq \frac 1n$. Prove that $(x_n)$ and $(y_n)$ converge and also that $\lim_{n \to \infty}x_n = \lim_{n \to \infty}y_n$.

This is part of a test of my first semester of undergraduate Calculus. I already gave it a shot, but I would like to know some other approaches to it.

What I already tried consisted basically in trying to prove that both sequences are bounded below and above - then, by the Monotone Convergence Theorem both sequences must converge. The rest was to prove that both sequences have the same limit from the given inequality.

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For $n\in \Bbb N$,
$$x_0\le x_n\le y_n\le y_0$$