Difference in probabilities of order $o(1)$

Question

Let $X_n, Y_n$ be random variables which satisfy $|X_n-Y_n|\leq C$ for some constant $C$. Let $f:\mathbb{N}\to\mathbb{R}$ satisfy $f(n)\to \infty$ for $n\to\infty$. Is the following equality true? $$\Pr\left\{|X_n|>f(n)\right\}=\Pr\left\{|Y_n|>f(n)\right\}+o(1).$$

If so, how to show it?

Answer 1

No, since we could have $X_n$ concentrated about $n-1$, $Y_n$ concentrated about $n+1$, and $f(n)=n$.