Question 31
The sequence $(a_n)_{n \ge 1}$ of real numbers is such that the sequence $(x_n)_{n \ge 1}$ defined by $x_n = \max\{a_n, a_{n+1}, a_{n+2}\}$ is convergent and the sequence $(y_n)_{n \ge 1}$ defined by $y_n = a_{n+1} - a_n$ has limit $0$. Prove that the sequence $(a_n)_n$ is convergent.