Question 12
Let $f: [0, 1] \to [0, 1]$ be an increasing function and let
$$
a_n = \int_0^1 \frac{1 + (f(x))^n}{1 + (f(x))^{n+1}} \, dx, \quad n = 1, 2, 3, \dots
$$
Prove that the sequence $(a_n)_{n \ge 1}$ is convergent and evaluate its limit.
$$
a_n = \int_0^1 \frac{1 + (f(x))^n}{1 + (f(x))^{n+1}} \, dx, \quad n = 1, 2, 3, \dots
$$
Prove that the sequence $(a_n)_{n \ge 1}$ is convergent and evaluate its limit.