By merge I assume you mean add them together where both are independent.
The usual way to prove this is through moment generating functions but you can also use the convolution result: they are both aspects of the same idea, but they have different uses depending on what needs to be done analytically and statistically.
Moment-generating function - Wikipedia, the free encyclopedia
List of convolutions of probability distributions - Wikipedia, the free encyclopedia
The proof from first principles is best done with the MGF to show that the products of two Poisson MGF's produce a Poisson MGF and since MGF unique represents a distribution, you're done.