Suppose that are i.d.d. to some distributions and are i.d.d. to .

If all the X's and Y's are indepedent to one aother, does that also mean that the sums is also independent to ?

So I'm trying to prove it by doing it striaght from the definition:

and try to get it to become

But I'm getting stuck on the left hand side, not sure how to break it down...

Any help, please? Thank you!