Category:Definitions/Limits Inferior of Set Sequences

From ProofWiki
Jump to navigation Jump to search

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

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

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

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