Let $x_1> 1$ and let $x_{n+1} := 2 – \displaystyle\frac{1}{x{_n}}$ for $n \in \mathbb{ N}$. Show that $(x_n)$ is bounded and monotone. Find the limit. I am confused on how to show that the sequence is increasing or decreasing without having a specific value for $x_1$. I think that I should use induction, but how do I define my base case?
[Math] Proving that a sequence is monotone and bounded
real-analysis
Best Answer
Hint. To show that the sequence is always increasing, the base case would be that $x_2>x_1$, that is, $$2-\frac{1}{x_1}>x_1\ .$$ Similarly, to show it is always decreasing, you would need to start with $$2-\frac{1}{x_1}<x_1\ .$$ By doing a little algebra you should be able to work out which of these is correct.