MathDB

Let

f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}

be a strictly increasing function. Prove that:
[*]There exists a decreasing sequence of positive real numbers,

(y_{n})_{n\in\mathbb{N}}

, converging to

0

, such that

y_{n}\leq2y_{f(n)}

, for all

n\in\mathbb{N}

. [*]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}\leq y_{n} \leq2y_{f(n)}

, for all

n\in\mathbb{N}

Problem Statement