I have a recurrent sequence defined as $a_1={1 \over 2}, a_2=1, a_n={1 \over 2}a_{n-1}+\sqrt{a_{n-2}}$. It seems to be increasing monotonically.
I assume that $a_n \to g$. If this is true, then $a_{n-1} \to g, a_{n-2} \to g$. Plugging this back to the equation gives me $g_1=0, g_2=4$. All the terms of the sequence are positive, so $g=4$ is the candidate for a limit.
I am able to show $4$ is an upper bound by induction. Now, I need to show that the sequence is in fact monotonic, and I'm done. The usual way is to write $a_{n+1}=f(a_n)$, where $f$ is some function which is obviously monotonic (so far I've worked with $f(x)=\sqrt{x}$), look at the first two terms of the sequence and then conclude by induction whether it is increasing or decreasing (or possibly stable). My problem is, I can't find an easy enough function of one variable. I have tried to show $a_{n+1}>a_{n}$ by induction but get stuck here as well. Any hints?
If: $$a_{n}\leq a_{n+1}\leq a_{n+2}$$ then also: $$a_{n+2}=\frac{1}{2}a_{n+1}+\sqrt{a_{n}}\leq\frac{1}{2}a_{n+2}+\sqrt{a_{n+1}}=a_{n+3}$$
So $$a_{n}\leq a_{n+1}\leq a_{n+2}\leq a_{n+3}$$
And the expansion can be continued.