The infinity of random variable 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 $[N < \infty]$?

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.
