Apply the definition of conditional probability:
.
So
Also we know so the non-negativity of the conditional probability of A given B is easy to show.
Now consider by definition is
by basic set theory,
since we assume mutual exclusivity
I am supposed to show that if P(B) is not equal to 0, then P(A|B) is greater than or equal to 0, P(B|B) = 1, and P(A1 union A2 union A3...|B) = P(A1|B) + P(A2|B) + P(A3|B) + ... for any sequence of mutually exculsive events A1, A2, A3...
It doesn't seem like it should be so hard... in fact, it makes perfect sense that P(B|B) = 1... but I really don't know how to show it. Please help!