TITLE: Exercise 1.6.3 from Alon & Spencer's *The Probabilistic Method*: prove that $Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1]$ for i.i.d. real RVs $X$ and $Y$
QUESTION [31 upvotes]: Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:

(A-S 1.6.3) Prove that for every two independent identically distributed real random variables $X$ and $Y$,
  $$Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1].$$

REPLY [7 votes]: You may read the paper "The 123 theorem and its extensions" by Noga Alon and Raphael Yuster.