Category:Definitions/Limits Superior of Set Sequences

From ProofWiki
Jump to navigation Jump to search

This category contains definitions related to Limits Superior of Set Sequences.
Related results can be found in Category:Limits Superior of Set Sequences.

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

Then the limit superior of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as:

\(\ds \limsup_{n \mathop \to \infty} E_n\) \(:=\) \(\ds \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n\)
\(\ds \) \(=\) \(\ds \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots\)