Wikipedia has an article that rigorously defines this concept using scaring symbols such as limsup, liminf, bigcap, and bigcup. However, I found a post on MO that explains the same idea intuitively. The keypoint is we can categorize every point ever appearing in the infinite sequence of sets $S_1,S_2,...$ into three partitions:
I) Present in the all but finitely many sets
II) Present only in finitely many sets
III) Present in and absent from the sets infinitely often
If partition III is nonempty then the limit of the sequence clearly doesn't exist. If the limit does exist, then it should contain exactly the points in partition I. The limit of a monotone set sequence always exist because every point in this sequence falls in the first two partitions. Hence for any sequence of sets $S_1,S_2,...$ the following sequences converge: $A_n\equiv \bigcup_{i=1}^n S_i$, $B_n\equiv \bigcup_{i=n}^\infty S_i$, $C_n\equiv \bigcap_{i=1}^n S_i$, $A_n\equiv \bigcap_{i=n}^\infty S_i.$
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