# Sets with infinite elements and intersection over sets

I am asked to state whether the following is true or if false to give a counterexample:

If $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots$ are all sets containing an infinite number of elements, then the intersection $$\bigcap_{k=1}^\infty A_k$$ is infinite as well.

I believe this statement to be false but I am not sure if the counterexample I have thought up makes sense. I said:

Let $A_n = \{m \in \mathbb{Z} | m> n\}$ for $n \in \mathbb{N}$. Would this be okay?

• Your counterexample is correct. You should probably state explicitly that the intersection is empty (and therefore not infinite). Sep 7, 2017 at 0:49
• @AndreasBlass Yes, I did mention it is empty and therefore not infinite.
– Sam
Sep 8, 2017 at 0:51

Hint: Consider $A_n = \left[-\dfrac1n, \dfrac1n \right] \subseteq \mathbb R$.
Not quite, because then you have a finite intersection, i.e. each $A_n$ has a finite amount of elements. How about trying $A_n = [n,\infty) \cap \mathbb{N}$? What is the intersection then?