Is the limit of this sequence of positive numbers equal to zero?

Let $\theta_1, \theta_2 >1$ and $c>0$. Let $\{a_k\}$ be a sequence of positive numbers such that for any $k\in\mathbb N$:

$$1. \quad 0 < a_0\le 1;$$

$$2.\quad a_{k+1}\le c^k(a_k^{\theta_1} +a_k^{\theta_2}).$$

Knowing these information, can one conclude something about the limit of $a_k$ as $k$ goes to $+\infty$?

I would say that the limit is $0$, because each $a_k\in (0, 1)$, but I do not know how to prove this.

Anyone could help?