Building $A_k$ such that $\displaystyle\bigcup_{k\in \mathbb{N}}A_{k} = \mathbb{R}$ and $\displaystyle\bigcap_{k\in \mathbb{N}}A_{k} = \mathbb{N}$ I wanted to know how I can find a sequence of sets according to a particular union and intersection of these sets? For example:

Build a sequence of sets such that
$$\bigcup_{k\in \mathbb{N}}A_{k} = \mathbb{R}$$
and
$$\bigcap_{k\in \mathbb{N}}A_{k} = \mathbb{N}$$

Is there a special method or just guess and try?
Thank you in advance.
Example 1

Take $A_1=\mathbb R$, $A_2=A_3=...=\mathbb N$, and we are done !

Example 2

A probably more interesting example, take for all $k$, $$A_k=\bigcup_{i=-\infty }^\infty \left[i-\frac{1}{k},i+\frac{1}{k}\right].$$
The case with $\mathbb R$/$\mathbb N$ is just an example, but the general case is just as easy.

The union consists of any element that is in just one of the $A_k$'s, so set $A_1$ to be that union.

All your $A_k$'s have to contain the intersection, so set the rest of the $A_k$'s to be the intersection. - And you're done.