Please someone help me to prove the following theorem: $$\mathbb{E}(X+Y)=\mathbb{E}(X)+\mathbb{E}(Y)$$ using the definition of expectation $\mathbb{E}(X)$
My proof: $$\mathbb{E}(X+Y) = \sum_{x} \sum_{y}(x+y)P_{XY}(x,y)$$ $$= \sum_{x} \sum_{y}xP_{XY}(x,y) + \sum_{x} \sum_{y}yP_{XY}(x,y)$$ $$= \sum_{x} xP_{X}(x) + \sum_{y}yP_{Y}(y)$$ $$=\mathbb{E}(X)+\mathbb{E}(Y)$$
Is this proof right?