I am trying to understand the limit, sup, and inf definitions of a sequence of sets.
Based on the answer to this: What does the supremum of a sequence of sets represent?
I think I can make an intuitive connection of $\sup_{n \in \mathbb{N}} A_{n}$ being the union. However, the intersection of $A_n$ as $\inf_{n \in \mathbb{N}} A_{n}$ is a bit confusing to me. Based on my understanding, for an increasing sequence of sets, $ A_1 \subset A_2 \subset A_3 \subset \dots$ and $\bigcup_{n=1}^{\infty}A_n = A$, I can assume $\inf_{n \in \mathbb{N}} A_{n} = A_1$. Is this correct?
But how can I define infimum for a decreasing sequence of sets without the condition $\bigcap_{n=1}^{\infty}A_n = A$, for example, the set $A_n$ continuously decreasing towards the null set as $n \rightarrow \infty$? Is this possible? I am tempting to say $\inf_{n \in \mathbb{N}} A_{n}$ would be $\varnothing$ in that case...or do we just say the limit does not exist?
Lastly, if the sequence of sets do not have a common intersection for $A_n$ for all n, would $\inf_{n \in \mathbb{N}} A_{n}$ be simply $\varnothing$ as well?
Consider $A_1 = [-1, 1]$, $A_2 = [-0.5, 0.5]$, so in general $A_n = [-1/n, 1/n]$. $0$ is in all of the sets so the intersection is not empty. So yes, if there is no intersection then the infimum would be the empty set.