Consider the sequence $U_{n+1} = 2 - \frac{1}{U_n}$, where $U_0 = 2$

Question

1. Show by induction sequence is decreasing.
2. Show by induction $U_n \ge 0$ for all $n \ge 0$

I was able to solve 1. and I am seriously embarrassed that I am unable to solve 2 by induction. I know it is true. I know the sequence converges to 1.

Any hint is appreciated. Thanks

Answer 1

Hint:

You can prove $U_n \ge 0$ by proving the stronger statement that $U_n \ge 1$. Note that if this is true for some $n \ge 0$, then you have

$$\frac{1}{U_n} \le 1 \implies -\frac{1}{U_n} \ge -1 \tag{1}\label{eq1A}$$

so you then also get

$$U_{n+1} = 2 - \frac{1}{U_n} \ge 2 - 1 = 1 \tag{2}\label{eq2A}$$