1

Assuming {$A_1, A_2, A_3,...$} is a sequence of events, I thought that the event for infinitely many occurring $A_i$'s to occur was $\bigcap_{i=1}^\infty A_i$ but it is actually $\bigcap_{m=1}^\infty \bigcup_{n=m}^\infty A_n$.

I still don't understand why my answer doesn't encompass infinitely many $A_i$'s to occur. I thought that if my expression/event were to occur, because infinitely many $A_i$'s intersect with each other, then infinitely many $A_i$'s would occur.

ultracookies
  • 11
  • 5
    If $A_1$ does not occur but the other events occur do you consider infinitely many of the events as occurring or not? – geetha290krm Jan 04 '24 at 06:32
  • @geetha290krm I'd say yes because despite one of the events not occurring, that doesn't change the fact that infinitely many of them still are. But I'm still failing to see where you're going with this even after reading Davide's comment and clicking the link. – ultracookies Jan 06 '24 at 00:37

1 Answers1

0

It is true that if $B:=\bigcap_{i=1}^\infty A_i$ is satisfied, then infinitely many of the $A_i$ occur.

However, this is not the only case where infinitely many of the $A_i$ occur. As pointed out by geetha290krm, if for example $A_1^c\cap \bigcap_{i=2}^\infty A_i$ occurs, then infinitely many of the $A_i$'s occur but not your even $B$.

You can check that $\bigcap_{m=1}^\infty \bigcup_{n=m}^\infty A_n$ occurs if and only if infinitely many of the $A_i$ occur, as discussed here.

Davide Giraudo
  • 172,925
  • I am still failing to understand, even after reading the linked post. Are you implying that $lim sup A_n$ is a more robust version than my solution? – ultracookies Jan 06 '24 at 00:41