Limit of sequence with discrete convolution $a_n = 1 - b * a$

Question

I have a sequence \begin{equation}

$a_n = 1 - \sum\limits_{i=1}^{n-1}b_na_{n-i}$ where $b_n\leq{c}^n$, and $c<1$

Based on multiple simulations with varying parameters, I think that the sequence does converge to a limit and is indeed a Cauchy sequence.

Unfortunately, I'm unsure how to outline the proof. I've read about Young's inequality for convolutions multiple times, but I haven't found a way to apply it to this problem. Would it make sense to try proving convergence of the sequence via stability criteria for the corresponding LTI system (Z-Transformation..)?

sample simulation

Any hint is appreciated