I need a some help with the following statement:

Let $\displaystyle Y_1,Y_2,...,Y_n$ be independent r.v., where $\displaystyle Y_i$ has a gamma distribution, with parameters $\displaystyle \alpha_i$ and $\displaystyle \beta$ (where $\displaystyle \beta$ is fixed and the values of $\displaystyle \alpha$ are different). Prove that $\displaystyle U=Y_1+Y_2+...+Y_n$ has a gamma distribution with parameters $\displaystyle \alpha_1+\alpha_2+...+\alpha_n$ and $\displaystyle \beta$

so far I have:

$\displaystyle U=Y_1+Y_2+...+Y_n$

$\displaystyle m_U(t) = m_{Y_1}(t)\times m_{Y_2}(t)\times ... \times m_{Y_n}(t)$

$\displaystyle m_U(t) = (1-\beta)^{-\alpha_1} \times (1-\beta)^{-\alpha_2} \times ... \times (1-\beta)^{-\alpha_n}$

$\displaystyle m_U(t)=\prod^{n}_{i=1} (1-\beta)^{-\alpha_i}= \$

at which point I don't know how to continue it.