Sequence with infinite number of zeros

My question reads:

Let’s call a sequence $(x_n)$ zero-heavy if there exists $M\in\mathbb{N}$ such that for all $N\in\mathbb{N}$ there exists $n$ satisfying $N\leq\ n\leq\ N + M$ where $x_n = 0$.

If a sequence is zero-heavy does it necessarily contain an infinite number of zeros?

Now, I have decided that this is true, but I am not too sure how to phrase my reasoning correctly. I do not necessarily want to write out a proof as I want to just argue with direct reasoning. I was thinking along the lines of well if we know it is zero dense, then we can always find a $0$ in between the interval given. Then because this happens for all $N$ in a way the intervals continue infinitely then so do the $0$'s.

I am not too sure if I am making too much sense but I do have a sense of what is going on in this problem. Any suggestions?