Set-builder notation on unique example

Question

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} $ }

Answer 1

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

Answer 2

The given set can be written as

{{x:x is a multiple of 1}, {x:x is a multiple of 2}....}

Therefore the set builder is

{X: X is a set containing the multiples of a given natural number}