$a \in [0,1]\hspace{2em}f(1) := 0,\hspace{1em} f(n + 1) := \frac{1}{2}(a+(f(n))^2),\hspace{2em} n \in \mathbb{N}$
Prove:
$f(n) ≤ 1-\sqrt{1-a}$
I assume that I'll need to convert this recursive function to a non-recursive one in order to prove the inequation.
I've already calculated the values for 2 to 4:
$\hspace{2em}f(2) = \frac{1}{2}a$
$\hspace{2em}f(3) = \frac{1}{8}a^2 + \frac{1}{2}a$
$\hspace{2em}f(4) = \frac{1}{128}a^4 + \frac{1}{16}a^3 + \frac{1}{8}a^2 + \frac{1}{2}a$
I can see the regularity, but I don't know how to express it with what I learned so far.