MathDB

Decide if the following statement is true or false: For every sequence

\{x_n\}_{n\in \mathbb{N}}

of non-negative real numbers, there exist sequences

\{a_n\}_{n\in\mathbb{N}}

and

\{b_n\}_{n\in\mathbb{N}}

of non-negative real numbers such that: (a)

x_n = a_n + b_n

for all

n

; (b)

a_1 + \cdots + a_n \le n

for infinitely many values of

n

; (c)

b_1 + \cdots + b_n \le n

for infinitely many values of

n

Problem Statement