For a sequence $\{x_n\}$ prove boundedness and find $\sup$ and $\inf$

Question

For the following sequence, defined by recursion as

\begin{align*} x_1 &= 0.5\\ x_{n+1}&=(2-x_n) \end{align*}

prove that the sequence is bounded and find its $\sup$ and $\inf$.

Answer 1

The key observation is that the sequence is simply $ \{ 0.5, 1.5, 0.5, 1.5, \cdots \} $

This is obviously bounded $ 0.5 \le x_k \le 1.5 $

$ \sup x_k = 1.5 $

$ \inf x_k = 0.5 $

The only thing that remains is to prove that the sequence is actually like that, we can prove that using a simple induction.

We are basically saying $ x_k = 1 + 0.5(-1)^k $, it is obviously true for $ k = 1 $.

Assuming it is true for $ k = N $, for $ k = N + 1 $, we have

$ x_{N + 1} = 2 - x_{N} = 2 - (1 + 0.5(-1)^{N}) = 1 - 0.5(-1)^{N} = 1 + 0.5(-1)^{N+1}$

Therefore by the principle of mathematical induction this is true for all natural numbers, the rest just simply follow!