# The probability of countably infinite independent sequence of events as a product of probabilities

My google-fu is failing, and I can't figure out how to approach this problem. Doing a grad level course on mathematical probability. We're given $A_n,n\ge 1$ is a (countably infinite) sequence of independent events. The only definition I've been able to dig up for this is that any finite subcollection is mutually independent, i.e. $P(\bigcap_{n=1}^kA_n)=\prod _{n=1}^kP(A_n)$.

I'm being asked to prove this holds for the whole infinite sequence, i.e. $P(\bigcap_{n=1}^\infty A_n)=\prod _{n=1}^\infty P(A_n)$

My classmates and I have no idea where to start at this one...

• Start with the equality you have (for finite $k$), and argue that the limit of each side as $k\to\infty$ is what appears in the equation you want to prove. – user147263 Sep 21 '14 at 21:18
• That's what I was thinking on the ride home right after I typed this, seemed almost too trivial. – Alan Sep 22 '14 at 21:11

Start with the equality you have (for finite $k$), and argue that the limit of each side as $k\to\infty$ is what appears in the equation you want to prove. This is how one normally obtains a statement for infinite families from a statement about finite families.
More specifically, $P(\bigcap_{n=1}^k A_n)\to P(\bigcap_{n=1}^\infty A_n)$ by the continuity of measure with respect to nested sequences, and $\prod _{n=1}^k P(A_n)\to \prod _{n=1}^\infty P(A_n)$ by the definition of an infinite product.