# algebra precalculus – Show that \$x_{n+1} = frac{2+x_n^2}{2x_n}\$ is a decreasing sequence.

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?