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I want to prove that the sequence $(x_n)_{n \ \in \ \mathbb{N}}$ defined below

$$\begin{cases} x_1 > 2 \\ x_{n+1}= \frac{1}{2}(x_n+\frac{2}{x_n}),\ n \in \ \mathbb{N}. \end{cases}$$

is monotonic.

I think the sequence is monotonically decreasing, however, I don't know how to prove it. I showed that $(x_n)_{n \ \in \ \mathbb{N}}$ is strictly positive in an attempt to show that

$$\frac{x_{n+1}}{x_n}<1, \forall \ n \in \mathbb{N},$$

and conclude that the sequence is monotonically decreasing.

Tiago Coelho
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  • It does! Thank you very much! – Tiago Coelho Apr 13 '24 at 22:37

1 Answers1

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To show $x_n$ is monotonically decreasing, it suffices to show $x_{n+1} -x_{n} < 0 $ for all $n \in \mathbb{N}$. First, we have $x_{n+1} -x_{n} = \frac{2}{x_n} - \frac{x_n}{2}$, which is less than $0$ if $x_n > 2$ for all $n$. For the later claim, we can show it by induction: we know $x_1>2$. Suppose $x_n>2$, we need to show $x_{n+1}>2$, which is not too hard by realizing that the function $f(x) = \frac{1}{2}(x + 2/x)$ is increasing in $x$ for $x>2$ and $f(2)>2$.

HSGLP
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