> [!definition] > > Let $E_1, E_2, \cdots$ be an [[Limit|infinite]] [[Sequence|sequence]] of [[Set|sets]] indexed by the [[Natural Numbers|natural numbers]]. Define the following sets: > $ > \underline{E} = \bigcup_{n = 1}^{\infty} \bigcap_{m = n}^{\infty} E_m \quad > \bar{E} = \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m > $ > > The limit $\limv{n}E_n$ exists if and only if $\underline{E} = \bar{E}$. $ \bigcap_{n = 1}^{\infty}E_n \subseteq \underline{E} \subseteq \bar{E} \subseteq \bigcup_{n = 1}^{\infty} $