Question 91
Consider an increasing continuous function $f : [0, 1] \to \mathbb{R}$ and define the sequence $(a_n)_{n \ge 1}$ by $a_n = \frac{1}{2^n} \sum_{k=1}^{2^n} f(\frac{k}{2^n})$, for all integers $n \ge 1$.
a) Prove that the sequence $(a_n)_{n \ge 1}$ is increasing.
b) Given that there exists $p \in \mathbb{N}^*$ such that $a_p = \int_0^1 f(x)dx$, prove that $f$ is a constant function.
a) Prove that the sequence $(a_n)_{n \ge 1}$ is increasing.
b) Given that there exists $p \in \mathbb{N}^*$ such that $a_p = \int_0^1 f(x)dx$, prove that $f$ is a constant function.