Question 88
Let $f: \mathbb{N} \to \mathbb{N}^*$ be a strictly increasing function. Prove that:
a) there exists a decreasing sequence of positive real numbers, $(y_n)_{n \in \mathbb{N}}$, converging to $0$, such that $y_n \le 2y_{f(n)}$, for all $n \in \mathbb{N}$;
b) if $(x_n)_{n \in \mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_n)_{n \in \mathbb{N}}$, converging to $0$, such that $x_n \le y_n \le 2y_{f(n)}$, for all $n \in \mathbb{N}$.
a) there exists a decreasing sequence of positive real numbers, $(y_n)_{n \in \mathbb{N}}$, converging to $0$, such that $y_n \le 2y_{f(n)}$, for all $n \in \mathbb{N}$;
b) if $(x_n)_{n \in \mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_n)_{n \in \mathbb{N}}$, converging to $0$, such that $x_n \le y_n \le 2y_{f(n)}$, for all $n \in \mathbb{N}$.