Question 35
Let $a$ be a real number in the open interval $(0, 1)$, let $n$ be a positive integer and let $f_n: \mathbb{R} \to \mathbb{R}$, $f_n(x) = x + x^2/n$. Show that
$$
\frac{a(1-a)n^2 + 2a^2n + a^3}{(1-a)^2n^2 + a(2-a)n + a^2} \lt \underbrace{(f_n \circ \dots \circ f_n)}_{n} (a) \lt \frac{an + a^2}{(1-a)n + a}.
$$
$$
\frac{a(1-a)n^2 + 2a^2n + a^3}{(1-a)^2n^2 + a(2-a)n + a^2} \lt \underbrace{(f_n \circ \dots \circ f_n)}_{n} (a) \lt \frac{an + a^2}{(1-a)n + a}.
$$