TITLE: Construct disjoint sequence of infinite subsets of $\mathbb{N}$ whose union is not $\mathbb{N}$
QUESTION [7 upvotes]: Possible to construct a disjoint sequence of infinite subsets of $\mathbb{N}$ whose union is not $\mathbb{N}$?

I am thinking this must be possible.
I've considered trying to define each sequence as the range of some function. I have to find a way to hop over natural numbers in some way to leave infinitely many left for each subsequent subset of $\mathbb{N}$. Thinking about just leaving out $1$ so then the union isn't $\mathbb{N}$. I haven't tried to do anything like this before. Any possible hints on how this can be done and I can think a little more to see if I can come up with something? If not I may just ask for an answer heh.
Thanks much for your time and input I appreciate it very much.

REPLY [10 votes]: Consider $A_n = \{ p_n^k : k \geq 1 \}$ where $p_n$ is the $n$th prime number.