Question 30
Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of positive real numbers such that for all $n \geq 1$,
$$
a_{n} \leq a_{2 n}+a_{2 n+1}
$$
Prove that there exists an $N \geq 1$ such that
$$
\sum_{n=1}^{N} a_{n}\gt 1
$$
$$
a_{n} \leq a_{2 n}+a_{2 n+1}
$$
Prove that there exists an $N \geq 1$ such that
$$
\sum_{n=1}^{N} a_{n}\gt 1
$$