In today's statistics class, we saw properties of the distribution function, i.e. defined by $F(x) = P(X\leq x)$ for a random variable $X$. One of these was:
$F(x)$ is right continuous.
The proof was:
Let $E_n$ be a decreasing sequence of events s.t. $\cap_{i=1}^{\infty} E_i = E$, with $E = \{X\leq x_0\}$ Then$$F(x_0) = P(X\leq x_0) = P(E) = P(\cap_i E_n) = \lim_i P(E_i) = F(x_0^{+}).$$
Surely I must be missing something since I don't see how she jumped from the third equality to the next one. Could you tell me why those equalities in the proof are true?
