Definition:Limit of Sets

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Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets.

Let the limit superior of $\Bbb S$ be equal to the limit inferior of $\Bbb S$.

Then the limit of $\Bbb S$, denoted $\ds \lim_{n \mathop \to \infty} E_n$, is defined as:

$\ds \lim_{n \mathop \to \infty} E_n := \limsup_{n \mathop \to \infty} E_n$

and so also:

$\ds \lim_{n \mathop \to \infty} E_n := \liminf_{n \mathop \to \infty} E_n$

and $\Bbb S$ converges to the limit.

Also see

  • Results about limits of sets can be found here.