# 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.

• Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Oct 8, 2017 at 14:40

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