We are given a sequence $(a_n)$ such that $0\lt a_n\leq a_{2n}+a_{2n+1}$ for all natural numbers $n$. Show that $\sum_{n=1}^\infty a_n$ diverges.
I attempted the question as follows. Consider, say, $$\begin{align} S_7 &= a_1+(a_2+a_3)+(a_4+a_5)+(a_6+a_7)\\ &\geq a_1+a_1+(a_2+a_3)\\ &\geq a_1+a_1+a_1 \end{align}$$
But I have no idea how to generalise this to show that the sequence diverges.