Let $\{P_n\}, n\in \mathbb{N}$ be the sequence prime numbers such that $P_1=2, P_2=3\dots$.
Define a new sequence $\{M_n\}$, $n\in \mathbb{N}$, such that $M_n=$Product of the digits of the $nth$ prime in its decimal representation.
Now the question is: Does there exist an $N\in \mathbb{N}$, such that $M_n=0$ $ \forall n\geq N$ ?
No. This 2016 paper by James Maynard says that there are infinitely many primes not containing the digit $0$. So $M_n \gt 0$ infinitely often.