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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.

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    Hint: Think about set theory. If $x$ is in that infinite union, then what? – 2017-01-23
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    Yes I think this hint and the more explicit answer User1006 gave both helped me at least with half of it. Does the other side of the proof, if x belongs to all events $ A_{2},A_{2}...$ except possibly a finite number of those events then x is in $\bigcup_{i=n}^\infty C_{n}$, need more effort? – 2017-01-23
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    To other side, just choose the max $m$ that $x\notin A_m$. Then for all $n>m, \:x\in C_n$ – 2017-01-23
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    Thank you is there any sort of resolved tag on stack exchange I should hit now? – 2017-01-23
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    Correcting the several typos in your formulas (title and body) would be a plus. – 2017-01-24

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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