Show that an outcome in S to the event $ \bigcup_{i=n}^\infty C_{n}$ if and only if it belongs to all the events $A_{2},A_{2}$... Show that an outcome in S belongs to the event $ \bigcup_{i=n}^\infty C_{n}$ where $C_{n}=\bigcap_{i=n}^\infty A_{i}$ if and only if it belongs to all the events $A_{2},A_{2}...$ except possibly a finite number of those events.
P.S. This is my first post so I probably did something wrong in posting this.

Suppose $\omega\in \bigcup_{i=n}^\infty C_{n}$. Then there is a $N>n$ that $\omega\in C_N$. Since $C_{N}=\bigcap_{i=N}^\infty A_{i}$, this means that for all $i\geqslant N, \:\omega\in A_i$. So if $\omega\notin A_i$, then $i<N$, i.e. there are only a finite number of those events $A_i$ that $\omega\notin A_i$ .
