Set-theoretic limit

Set-theoretic limit: 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 "x_i" equal 1 if "x" is in "A_i" and 0 otherwise. If the limit as "i" goes to infinity of "x_i" exists for all "x", define: $lim_{i
ightarrow infty} A_i = { x : lim_{i
ightarrow infty} x_i = 1 }.$
*Using union and intersection, define: $liminf_{i
ightarrow infty} A_i = igcup_i igcap_{j geq i} A_j$ :and: $limsup_{i
ightarrow infty} A_i = igcap_i igcup_{j geq i} A_j$ .:If these two sets are equal, then either gives the set-theoretic limit of the sequence.

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Set-theoretic limit

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