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In mathematics, the limit of a sequence of sets A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } is a set whose elements are determined by the sequence in either of two equivalent ways: by upper and lower bounds on the sequence that converge monotonically to the same set and by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual.
More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical.. Such set limits are essential in measure theory and probability.
It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of x = lim k → ∞ x k , {\displaystyle x=\lim _{k\to \infty }x_{k},} where each x k {\displaystyle x_{k}} is in some A n k . {\displaystyle A_{n_{k}}.} This is only true if convergence is determined by the discrete metric. This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below.