Proof: Product of Expectation of two independent random variables!

Consider two independent Random variables A, and B, now I know that, E[A+B] = E[A] + E[B], E[AB] = E[A] * E[B].

I am looking for a prove of these properties, I am successful in proving the first one, but I am unable to prove the 2nd property.

Can anyone throw some guideline, or a starting point for the second proof?

Regards,

Solution 1:

In the finite case with $E[A]=\sum_j p_j a_j$ (where $p_j=P(A=a_j)$) and $E[B]=\sum_k q_k b_k$ (where $q_k=P(B=b_k)$), we have $$ E[AB]=\sum_{j,k} p_jq_k a_jb_k=\sum_j p_ja_j\cdot\sum_kq_kb_k=E[A]E[B]$$ (where by independence $p_jq_k=P(A=a_j)P(B=b_k)=P(A=a_j,B=b_k)$).