To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
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 $Y$