Show the sequence $\{a_{n}\}_{n=1}^{\infty}$ is unbounded

By the limit hypothesis, for $n$ large enough, \begin{equation} \frac{a_n}{a_{n+1} + a_{n+2}} < \frac{1}{4}. \end{equation} Therefore, either $a_{n+1} > 2 a_n$ or $a_{n+2} > 2 a_n$. Using that, we can find a subsequence $a_{i_0}, a_{i_1}, \ldots$ such that $a_{i_n} \ge 2^n a_{i_0}$ (by choosing $i_{n+1}$ to be either $i_n + 1$ or $i_n + 2$). That easily implies that the subsequence is unbounded, and therefore the original sequence is also unbounded.