Suppose non-negative sequence $\{a_n\}$ satisfies the following relation: $$a_{n+1} - 2a_n \leq b_n+c_n a_n + d_n a_n^{\frac{p-1}{p}},$$ where $b_n, c_n , d_n$ are the sequences converging to $0$ and $p > 1$ be any real number.
Now I am trying to prove the convegence of above sequence $\{a_n\}$. Does it converge towards $0$ or what condition should we apply on it so that this sequence converges towards $0$?
Any help thanks.