# Set-builder notation on unique example

I understand the fundamentals of set-builder notation, however, a more sophisticated problem with nested sets is stumping me. I'd really like some constructive criticism on how I approach this problem and even some wisdom for future problems. As an example, I'll be using set-builder notation on:

A set with infinitely many elements with each element being a set that itself
has infinitely elements such that:

{{1,2,3,...}, {2,4,6,...}, {3,6,9,...}, ...}.


My thought process:

• I need two variables to define two sets to work with
• One variable, set 1, needs to be the empty set
• The other variable, set 2, needs to be the set of natural numbers
• The resulting set is set 2 added to set 1 infinitely many times

My notation:

• {x $\in\emptyset$: x = {x + y} where y $\in\mathbb{N}$ }
• $\{x \in \emptyset : [\text{anything here}] \}$ is always $\emptyset$. Can you write each of the "inner" sets using set builder notation? Can you write a "generic" formula for an inner set using set builder notation (it will probably involve a parameter, say $k$)? If you can, let's call the inner set $S_k$ for a moment. Then your set is $\{S_k : k \in \Bbb N\}$, and just replace $S_k$ with its set-builder notation. – pjs36 Apr 12 '17 at 4:27

$\Bigl\{ \{n\cdot k \mid k\in\mathbb{N}\} ~~~~ {\large\mid} ~~~~n\in\mathbb{N} \Bigr\}$