Prove the intersection of events to be 1 $B_n$, $n$ from $1$ to infinity is countably infinite sequence of events and each event has probability $1$. How do I formally prove that the probability of intersection of $B_n$ from $n = 1$ to infinity is also $1$. Intuitively I know because $P[B_i] = 1$, so all events are equivalent to the total sample space, and so does its intersection. But how to prove it in formal way?

Hint: The probability of at least one of the events not happening is at most the sum of the probabilities of each event not happening, which is zero because there are only countably many of them each with probability zero of not happening.
