Consider a sequence $(a_i)_{i\in\mathbb{N}}$ of non-negative real numbers and $(b_i)_{i\in\mathbb{N}}$ a positive sequence tending to zero as $i\to\infty$. Furthermore, suppose that there is a $c>0$ such that $$ a_i\geq b_i + ca_{i-1}.$$
I try to get a lower bound on $a_i$ for each $i\in\mathbb{N}$ and hoped to find a Gronwall type lemma but did not succeed. What quite obviously holds is $$a_i\geq b_i+c(b_{i-1} +c(b_{i-2}+c(...)))$$
and what also follows directly is the lower bound $a_i\geq c^i$ by omitting all the $b_i$. But I want to find a lower bound that depends on the $b_i$. How can I compute the extended right hand side or is there some lemma I could use?