# What is the expansion of $\cap ^\infty _{m=1}\cup^\infty _{n=m}X_n$?

It's the first time I'm seeing it and I'm having trouble constructing it inductively. I am guessing it's one of the two, but I am probably wrong:

$$X_1 \cap (X_1 \cup X_2)\cap (X_1\cup X_2\cup X_3)\cap...$$ $$X_1 \cap (X_1 \cup X_2)\cap ((X_1 \cap (X_1 \cup X_2))\cup X_4)...$$

$$\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right)$$ is equal to:
$$(X_1\cup X_2\cup X_3\cup \dots) \cap (X_2\cup X_3\cup X_4\cup \dots) \cap (X_3\cup X_4\cup \dots) \cap \dots$$
It so happens that in the event that $$X_n$$ "converges" to a specific set $$X$$ then $$\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right) = X$$ (for example in the event that we are talking about nested sets such as when $$X_1\subseteq X_2\subseteq X_3\subseteq\dots$$)
What you asked can also be found in the domain of probabilities:Consider $$X_n$$ to be a sequence of events and then $$\bigcap_{m=1}^\infty \left(\bigcup_{n=m}^\infty X_n\right)$$ will be the $$limsupX_n$$ If you study it there,you may be able to understand it better.
I don’t think it’s either of your suggestions, but it’s the set of all elements that appear in infinitely many $$X_n$$.