SEEMOUS 2023 Problem 4

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$ . [list=i] []For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$ , solution of the equation $f(x)=x^n$ . Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$ . []Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$ . For any $x\in\mathbb{R}$ we define $$F(x)=\int_x^1f(t) \ dt.$$Let $\alpha{}$ be a real number. Study the convergence of the series $$\sum_{n=1}^\infty F(a_n)^\alpha.$$ [/list]