Define a sequence \(\displaystyle \{x_n\}_{n=1}^{\infty}\) by \(\displaystyle x_1 = 1, x_2 = \frac{1}{2}\) and\(\displaystyle x_n = \frac{2x_{n-1} + x_{n-2}}{4} \)for \(\displaystyle n \ge 3.\) Use the Monotone Convergence Theorem to show that \(\displaystyle \{x_n\}_{n=1}^{\infty}\) converges and find its limit.

Ok it is not too hard to prove by induction that \(\displaystyle 0 \le x_n \le 1\). But I am having trouble proving the sequence is decreasing. I have

\(\displaystyle x_{n+1} - x_{n} = \frac{2x_n + x_{n-1}}{4} - x_n = \frac {x_{n-1} - 2x_n}{4}\), but I have no guarantee that this is less than or equal to zero!

Any help would be appreciated!