Show that n →∞lim sup En={x ∈S: x ∈En for infinitely many n} and that lim infn →∞ En={x ∈S: x

step 1 So, Lim sub -E -K is the intersection for N1 of the Union for K -N of E -K. And so by the Morgan

Show that n →∞lim sup En={x ∈S: x ∈En for infinitely many ...

Show that
$$
\underset{n \rightarrow \infty}{\limsup } E_n=\left\{x \in S: x \in E_n \text { for infinitely many } n\right\}
$$
and that
$$
\liminf _{n \rightarrow \infty} E_n=\left\{x \in S: x \in E_n \text { for all but finitely many } n\right\} .
$$

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Key Concepts

set operations on sequences

Understanding the use of unions and intersections is crucial when dealing with sequences of sets. The limsup is formulated as an intersection of unions, while the liminf is formulated as a union of intersections. Mastery of these operations enables one to rigorously define and analyze the limit behavior of collections of sets.

limsup of a sequence of sets

This concept refers to the set of all points that appear in infinitely many sets in a given sequence. It is constructed by taking the intersection over all indices of the union of sets from that index onward. In other words, a point is in the limsup if no matter how far you go along the sequence, there is always some later set that contains that point. This idea is fundamental in discussions about convergence and limit behavior in analysis and probability.

liminf of a sequence of sets

This concept represents the set of all points that eventually appear in every set of the sequence, except possibly for a finite number of exceptions. It is defined by taking the union over all indices of the intersection of sets from that index onward. Essentially, a point is in the liminf if there exists an index after which the point is included in every set. This notion captures a stronger form of convergence compared to the limsup.

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