By induction, it suffices to prove it for two random variables of parameters and of parameters .

The probability density function of the sum of two independent random variables (with densities) is the convolution of their density functions: in your case, this is:

.

There are a few simplifications, and you get:

(after the change of variable ). Call the last integral . Then we have found that the density function of is . This is a multiple of the density function of a Gamma distribution with parameters , and this is a density function, hence thisisthe density function of a Gamma distribution with parameters .

As a consequence this shows as well that , which you already knew, perhaps.

Why is the convolution involved? This is because, for every positive measurable function , we have, using Fubini:

,

and because these integrals characterize the distribution of .