Need help understanding the logic behind the intersection of infinite sets. As the post topic states, I need some help understanding the logic behind the intersection of infinite sets. I will take this time to thank anyone who reaches out with a comment or answer.
The following example has been taken from a real analysis book I've purchased.
Say the following sets exist,
$$ A_1 = N = \{1, 2, 3, ...\}\\
A_2 = \{2, 3, 4, ...\}\\
A_3 = \{3, 4, 5, ... \}\\
A_n = \{n, n+1, n+2, ...\}$$
Now, i'm okay with this. $$\bigcup_{n=1}^{\infty}A_{n} = A_1$$
This is what I can't seem to grasp.
$$\bigcap_{n=1}^{\infty}A_{n} = \emptyset$$
I can't make a mental picture of this since we have literally an infinite number of sets. I can see that $A_1 \cap A_2 = A_2$ applying that logic to each of these sets, which are by definition "infinite" there will always be some set $A_{n+1}$ for every $A_{n}$ So how can this ever be the empty set?

If some number $k$ is in this intersection, then it is in all of the sets $A_n$. But $k$ is definitely not in $A_{k+1}$. Therefore no such $k$ exists; the intersection is empty.
The key is that for something to be in an intersection, it must be in each and every one of the intersecting sets.
