How to prove that the moment generating function of the sum of two independent variables are multiplication of their moment generaing functions?

Question

How to prove that if $X \text{ and } Y \text{ are independent and } Z=X+Y \text{ and their mgf is } M_X \text{ and } M_Y \text{ then } M_Z=M_X \cdot M_Y?$

Answer 1

$$\begin{align}M_Z(t)&:=\Bbb E\exp itZ\\&=\Bbb E\exp itX\exp itY\\&=\Bbb E\exp itX\Bbb E\exp itY\\&=M_X(t)M_Y(t)\end{align}$$uses the definition of an MGF, the definition of $Z$, independence, and the definition of MGFs again.