Question 145
A sequence $(u_n)_{n=0}^{\infty}$ is defined recursively as follows: $u_0 = 2$ and
$$
u_n = \frac{2u_{n-1} + 1}{u_{n-1} + 2}, \text{ for } n = 1, 2, 3, \dots$$
So, for example, $u_1 = \frac{5}{4}$, $u_2 = \frac{14}{13}$. Prove that for all positive integers $n$,
$$1 \lt u_n \lt 1 + \frac{2}{3^n}.$$
$$
u_n = \frac{2u_{n-1} + 1}{u_{n-1} + 2}, \text{ for } n = 1, 2, 3, \dots$$
So, for example, $u_1 = \frac{5}{4}$, $u_2 = \frac{14}{13}$. Prove that for all positive integers $n$,
$$1 \lt u_n \lt 1 + \frac{2}{3^n}.$$