Set-theoretic limit

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In mathematics, the limit of a sequence of sets A1, A2, ... is a set whose elements are determined by the sequence in either of two equivalent ways:

  • Using indicator variables, let xi equal 1 if x is in Ai and 0 otherwise. If the limit as i goes to infinity of xi exists for all x, define
\lim_{i \rightarrow \infty} A_i = \{ x : \lim_{i \rightarrow \infty} x_i = 1 \}.
\liminf_{i \rightarrow \infty} A_i = \bigcup_i \bigcap_{j \geq i} A_j
and
\limsup_{i \rightarrow \infty} A_i = \bigcap_i \bigcup_{j \geq i} A_j.
If these two sets are equal, then either gives the set-theoretic limit of the sequence.
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