Question 55
Determine whether the following statement is true or false:
"For every sequence $x_{1}, x_{2}, x_{3}, \ldots$ of real numbers greater than or equal to zero there exist two sequences $a_{1}, a_{2}, a_{3}, \ldots$ and $b_{1}, b_{2}, b_{3}, \ldots$ of real numbers greater than or equal to zero such that
- $x_{n}=a_{n}+b_{n}$ for each $n$;
- $a_{1}+\ldots+a_{n} \leq n$ for infinite values of $n$;
- $b_{1}+\ldots+b_{n} \leq n$ for infinite values of $n$, possibly different from the previous ones."
"For every sequence $x_{1}, x_{2}, x_{3}, \ldots$ of real numbers greater than or equal to zero there exist two sequences $a_{1}, a_{2}, a_{3}, \ldots$ and $b_{1}, b_{2}, b_{3}, \ldots$ of real numbers greater than or equal to zero such that
- $x_{n}=a_{n}+b_{n}$ for each $n$;
- $a_{1}+\ldots+a_{n} \leq n$ for infinite values of $n$;
- $b_{1}+\ldots+b_{n} \leq n$ for infinite values of $n$, possibly different from the previous ones."