Let $x_n$ be defined as:

$$

begin{cases}

x_{n+1} = frac{2+x_n^2}{2x_n} \

nin mathbb N \

x_1 = 4

end{cases}

$$

Show that $x_n$ is a decreasing sequence.

I’m having a hard time with the sequence above. I’ve started with assuming that $x_{n+1} < x_n$. Now having that in mind we may inspect the following inequality:

$$

x < frac{2+x^2}{2x} iff 2x^2 < 2+x^2 iff x^2 < 2

$$

The inequality doesn’t show what’s needed but $sqrt2$ seems to be a point to which the sequence converges. I’ve also tried calculations with various initial conditions for $x_1$ and it looks like for all $x_1 > 0$ the sequence converges to $sqrt2$ while for $x_1 < 0$ it converges to $-sqrt2$.

Finding a closed form seems to not be an options since this recurrence is non-linear and i don’t think it has a closed form.

What would be a formal way to show that $x_n$ is decreasing?