Triangular inequality for expected value

Question

Is it true that if $X$ and $Y$ are random variables, then $E\left|X+Y\right|\leq E\left|X\right| + E\left|Y\right|$?

Is it required that $X$ and $Y$ are independent? How to prove it?

Answer 1

Yes it is true, and does not require $X$ and $Y$ to be independent. It follows (i) from the regular triangle inequality, which says that $|X+Y|\le |X|+|Y|$, (ii) from the monotonicity of expectation, which says that whenever $Z\le W$, we have $EZ\le EW$, and (iii) from the linearity of expectation, $E(Z+W)=EZ+EW$.