Question

Find a sequence of nested open intervals such that
A1 is a superset of A2
A2 is a superset of A3
A3 is a superset of A4...

Basically, the size of each set is like:
A_1 > A_2 > A_3 > A_4...

and the union of all of these subsets A_n, as n goes to infinity, is a finite set.

Answer 1

Failed Attempt 1:
A_n = (5 - 1/n , 5 + 1/n)
A_1 = (5 - 1/1, 6 - 1/1)
A_2 = (5 - 1/2, 5 + 1/2)
A_3 = (5 - 1/3, 5 + 1/3)
A_inf = (5,5) = empty set.

Answer 2

Failed Attempt 2:
A_n = (5 - 1/n , 6 + 1/n)
A_1 = (5 - 1/1, 6 - 1/1)
A_2 = (5 - 1/2, 6 + 1/2)
A_3 = (5 - 1/3, 6 + 1/3)
A_inf = (5,6) = infinite set

Answer 3

I have concluded that this is impossible.
The union of infinite number of sets can only be finite if it converges to a single digit, but since it's an open-interval, the single-digit will not be included in the set. So it will be empty.