# Why are these sets equal? From Morris H. DeGroot and Mark J. Schervish: Probability and Statistics, theorem 1.5.2.

"Proof. Consider the infinite sequence of events $$A_1, A_2, \dots,$$ in which $$A_1, \dots, A_n$$ are the $$n$$ given disjoint events and $$A_i$$ = $$\emptyset$$ for $$i > n$$. Then the events in this infinite sequence are disjoint and $$\bigcup\limits_{i=1}^{\infty} A_i$$ = $$\bigcup\limits_{i=1}^{n} A_i$$."

My question is why are the sets $$\bigcup\limits_{i=1}^{\infty} A_i$$ and $$\bigcup\limits_{i=1}^{n} A_i$$ equal here?

I'm confused because, suppose $$A_1$$ = {$$a_1$$}, $$A_2$$ = {$$a_2$$}, and $$A_3$$ = {$$a_3$$}. Then...

• $$\bigcup\limits_{i=1}^{\infty} A_i$$ = {$$a_1$$, $$a_2$$, $$\emptyset$$}

• $$\bigcup\limits_{i=1}^{n} A_i$$ = {$$a_1$$, $$a_2$$}

So it doesn't make sense to me that these sets are equal...

• The assumption is that $A_i=\emptyset$, not that $A_i=\{\emptyset\}$. – Lord Shark the Unknown Dec 29 '18 at 11:00
• If all the $A_i$ are empty, for $i > n$, then "adding" them to $\bigcup_{i=1}^n A_i$ does not add new elements. – Mauro ALLEGRANZA Dec 29 '18 at 11:00
• Oh that makes sense @LordSharktheUnknown, thank you! – Andres Kiani Dec 29 '18 at 11:01

You can see it this way. Take any element $$x$$ from $$\bigcup\limits_{i = 1}^{\infty} A_i$$. Then, it is in one of the $$A_i$$ for some $$i$$. Since $$A_i = \emptyset$$ for $$i > n$$, that $$i$$ has to be $$\leq n$$. Hence, $$x \in \bigcup\limits_{i = 1}^n A_i$$. Therefore, $$\bigcup\limits_{i = 1}^{\infty} A_i \subseteq \bigcup\limits_{i = 1}^n A_i$$.