## Definitions of Probability Measure Proof

Hi, I'm wondering if someone can help me with the following questions relating to the following definitions:

Given sample space S with event space A probability measure is defined as:
1) Non-Negative Function, P: A-->[0, infinity)
2) Normed, P(S)=1
3) Sigma-Additive, A = A1+...+An+...
2) Given that An--->A and Bn--->B prove that An*Bn--->AB and Complement(An)--->Complement(A) and $\displaystyle A_n \bigcup B_n --> A \bigcup B$
3) We say (6) P is sequentially continuous at 0 iff $\displaystyle A_n \downarrow \varnothing => P(A_n) \downarrow 0$