At first glance the following question:
Evaluate the limit of: $$1,\sqrt{a},\sqrt{\frac{a}{\sqrt{a}}},\sqrt{\frac{a}{\sqrt{\frac{a}{\sqrt{a}}}}},\cdots$$Where $a$ is a positive real.
Is trivial.
One assumes the sequence (call it $a_n$) approaches a fixed point $\alpha$, and then: $$\alpha=\sqrt{\frac{a}{\alpha}}\implies\alpha^3=a$$
More formally, the bog standard fixed point theorem gives that the sequence will converge, if $f(x)=\sqrt{\frac{a}{x}}$, when:
$$1\gt|f’(x)|=\frac{a}{2x^2}\sqrt{\frac{x}{a}}=\frac{\sqrt{ax}}{2x^2}$$
That is, when $x\gt\sqrt[3]{a/4}$.
Except, experimentally this is not the case. Experimentally the sequence tends to always converge for $a\gt0$. This is because after not so very long it tends toward the region of convergence described above, and after this of course it is guaranteed to converge. One possible reason related to the specific function is that its derivative is always increasing from below the region, and eventually will pass the boundary $-1$.
My question is, how do I know formally that it always ends up in the attraction basin (I think I’m using that word right)? I am not actually familiar with any fixed point theorems other than Banach’s contraction mapping principle, and I am dimly aware that the theory of such dynamics is quite developed. With what tools can one tackle this? I’m sure this particular sequence is more elementary, but in general how can we do better than the contraction mapping principle?