# Limits of Probabilities

• September 2nd 2009, 12:01 PM
azdang
Limits of Probabilities
Alright, another, hopefully quick, probability question.

Let ( $A_n$) and ( $B_n$) be in A with $A_n$-->A and $B_n$-->B, with P(B)>0 and P( $B_n$)>0. Show that:

a) $lim_{n->\infty}P(A_n|B)$=P(A|B)

b) $lim_{n->\infty}P(A|B_n)$=P(A|B)

c) $lim_{n->\infty}P(A_n|B_n)$=P(A|B)

So, I know that:

$P(A_n|B)$= $\frac{P(B|A_n)P(A_n)}{P(B)}$= $\frac{P(B \cap A_n)}{P(B)}$

$P(A|B_n)$= $\frac{P(B_n|A)P(A)}{P(B_n)}$= $\frac{P(B_n \cap A)}{P(B_n)}$

$P(A_n|B_n)$= $\frac{P(B_n|A_n)P(A_n)}{P(B_n)}$= $\frac{P(B_n \cap A_n)}{P(B_n)}$

$P(A|B)$= $\frac{P(B|A)P(A)}{P(B)}$= $\frac{P(B \cap A)}{P(B)}$

My problem really is how to show the limit of $P(B|A_n)$, $P(B_n|A)$, and $P(B_n|A_n)$ is $P(B|A)$. They seem obvious, but I can't figure out how to show it.