This exercise involves a well-known inequality known as the triangle inequality (a special case of Minkowski's Inequality).

Question:

This exercise involves a well-known inequality known as the triangle inequality (a special case of Minkowski's Inequality).
(a) Prove (without using Minkowski's Inequality) that for any numbers a and b
|a + 6| < |a| + |6|.
(b) Use part (a) to establish that for any random variables X and Y with finite expectations,
E|X + Y| < E|X| + E|Y|.