Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Question

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$, show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X$ and $Y$ are independent and continuous. I am confused how to proceed from here (see picture). Thank you.

My approach:

Answer 1

Your last line should have been $$m_{X+Y}(t) = \int_{-\infty}^\infty e^{ts} f_{X+Y}(s) \mathop{ds}= \int_{-\infty}^\infty e^{ts} \int_{-\infty}^\infty f_X(s-y) f_Y(y) \mathop{dy} \mathop{ds}.$$ Making the change of variables $s=x+y$ gives you the answer.