Suppose $\displaystyle \mathfrak{R} $ is a $\displaystyle \sigma $-ring. Suppose $\displaystyle E_n \in \mathfrak{R} $. How would we show that $\displaystyle \bigcap_{n=1}^{\infty} E_n \in \mathfrak{R} $?

We know that $\displaystyle \bigcup_{n=1}^{\infty} E_n = \bigcap_{n=1}^{\infty} E_{n}^{c} \in \mathfrak{R} $. Likewise, how would we show that $\displaystyle \overline{\lim} \ E_n \in \mathfrak{R} $ and $\displaystyle \underline{\lim} \ E_n \in \mathfrak{R} $? The former is the set of points that are in infinitely many $\displaystyle E_n $. The latter is the set of points that are in all but a finite number of $\displaystyle E_n $.