Let $(X_n)_{n\in\mathbb{N}}$ and $X$ be real valued random variables on a measure space with measure $\mu$ and let $X_n\rightarrow X$ $\mu$-almost surely. I was wondering if for every $\varepsilon >0$ there exists some $N\in\mathbb{N}$ such that $$\mu(|X_n-X|>\varepsilon)<\varepsilon$$
Here is my proof for this little proposition and I am unsure if it is correct:
Let $A_n:=\{|X_n-X|>\varepsilon\}$ then because the convergence is almost surely we have $\bigcap_{n\in\mathbb{N}}A_n\searrow N$ for some $\mu$-nullset $N$. By uppercontinuity of the measure it follows that $\lim_{n\to\infty}\mu(A_n)=\mu(N)=0$. In other words $$\forall\varepsilon' >0\,\exists N\in\mathbb{N}\,\forall n\ge N\colon |\mu(A_n)-\mu(N)|<\varepsilon'\\ \Leftrightarrow |\mu(A_n)|<\varepsilon'$$
and in particular for $\varepsilon'=\varepsilon$.