Comparing two events

Question

Let $X_n$ be a sequence of real random variables, $X$ a real random variable and $\varepsilon > 0$

Let $A_n$ be the event $ \{ | X_n - X | \geq \varepsilon \} $ and $B_n$ the event $ \{ \sup_{k \geq n} | X_k - X | \geq \varepsilon \} $

I was wondering if we can compare $B_n$ and $ \lim \sup A_n $

Thank you for your answer,

Answer 1

$$\bigcup_{m\geq n}\{|X_m-X|\geq \varepsilon \}\subset B_n$$ therefore $\limsup_{n\to \infty }A_n\subset B_n.$