Nested Interval Property and the intersection of infinite sequences

There is a theorem which generalizes the nested interval property:

Theorem: (Nested Compact Sets Property) Let $K_1\supseteq K_2\supseteq K_3\dots$ be a nested sequence of nonepmty compact sets in a Hausdorff topological space. Then $\bigcap_n K_n$ is nonempty.

Proof: If $\bigcap_n K_n=\varnothing$ , then $\{K_1\setminus K_n\}_{n\ge 2}$ is an open cover of $K_1$ with no finite subcover. $\square$

However, the assumption of compactness is necessary. In a general topological space, a nested sequence of sets $A_n$ can have null intersection. Your sets $A_n$ are a typical counter-example. Note that the $A_n$ are not compact.

In summary, the commonality to your two examples is nested nonempty sets, and the difference is compactness.

First of all note that both $\{A_n\}$ and $\{I_n\}$ are countable so this is not the problem.

The difference is in that $A_n$ is not a bounded closed interval, so we can not apply the nested interval theorem.

The nested interval theorem is about nested bounded closed intervals not arbitrary sets.