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The problem is:

For infinite independent Bernoulli trials, prove that the total number of successful trials $N$ have the following property:

$$ [N < \infty] = \bigcup\limits_{n=1}^{\infty}\,[N \le n] $$

Actually this is just part of bigger problem in a book, and the equation is given as an obvious fact and as a hint without any explanation. What does the equation exactly mean? I guess square brace means set, but what's the definition of $[N < \infty]$?

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@Didier, I think I finally got the idea! Can you turn this into an answer so I can mark it as correct? – ablmf Sep 11 '11 at 21:46
up vote 1 down vote accepted

Forget everything except that $N$ is a function from $\Omega$ to $\mathbb R^+$. Then $[N<\infty]$ is the set of $\omega$ in $\Omega$ such that $N(\omega)$ is finite and $[N\le n]$ is the set of $\omega$ in $\Omega$ such that $N(\omega)\le n$.

Hence $[N\le n]\subseteq[N<\infty]$ for every $n$. For the other inclusion, note that $N(\omega)$ finite implies there exists $n$ such that $N(\omega)\le n$. Hence the equality.

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$N$ is a random variable, the total number of successes. $[N < \infty]$ is the event that the total number of successes is finite. The equation says that $N$ is finite if and only if $N$ is at most $n$ for some positive integer $n$.

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