grandes-ecoles 2025 Q15

grandes-ecoles · France · x-ens-maths__psi Proof Proof by Induction or Recursive Construction
Let $c > 0$, and let $\left(a_n\right)_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that $a_{n+1} \leq a_n - c(a_n)^2$ for all $n \in \mathbb{N}$. Show $a_n \leq a_0/(1 + nca_0)$ for all $n \in \mathbb{N}$. Hint: adapt the reasoning from question 10.c)
Let $c > 0$, and let $\left(a_n\right)_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that $a_{n+1} \leq a_n - c(a_n)^2$ for all $n \in \mathbb{N}$. Show $a_n \leq a_0/(1 + nca_0)$ for all $n \in \mathbb{N}$.\\
Hint: adapt the reasoning from question 10.c)