Let $\{b_n\}_{n\geq0}$ be a sequence of numbers such that $b_nb_{n+1}=0$, and define $$a_n:=\sum_{k=0}^n(-1)^{n-k}\binom{n}{k}b_k.$$ If $\lim_{n\to\infty}a_n=0$, can we conclude that $b_n=0$ for all $n$?
I have tried many special $\{b_n\}$ but neither of them makes $\{a_n\}$ convergent. I think it’s a nontrivial problem.
More generally, if $\{b_n\}_{n\geq0}$ is a sequence with infinitely many zeros and $\lim_{n\to\infty}a_n=0$, can we still conclude that $b_n=0$ for all $n$?