Answer:
If the intersection is finite the statement is true, but if the intersection is infinite the statement is false.
Step-by-step explanation:
From the statement of the problem I am not sure if the intersection is finite or infinite. Then, I will study both cases.
Let us consider first the finite case:
. Because the condition A1 ⊇ A2 ⊇ A3 ⊇ A4 ... we can deduce that the set
is a subset of each set
with
. Thus,
.
Therefore, as
is infinite, the intersection is infinite.
Now, if we consider the infinite intersection, i.e.
the reasoning is slightly different. Take the sets
(this is, the open interval between 0 and
.)
Notice that (0,1) ⊇ (0,1/2) ⊇ (0, 1/3) ⊇(0,1/4) ⊇...So, the hypothesis of the problem are fulfilled. But,

In order to prove the above statement, choose a real number
between 0 and 1. Notice that, no matter how small
is, there is a natural number
such that
. Then, the number
is not in any interval
with
. Therefore,
is not in the set [tex]\cap_{k=1}^{\infty}(0,1/k)[\tex].