I'm preparing for an exam and I came across this problem from old exams. I'm really clueless on how to solve it.
Consider a sequence of random variables $\{X_n\}_{n=1} ^\infty$ defined on the probability space $([0,1],B[0,1],\lambda)$ where $\lambda$ is Lebesgue measure. Define $X_n(\omega) = 1_{[1/2-(2n)^{-1}, 1/2+(2n)^{-1} ]}(\omega)$ for all $\omega \in [0,1]$. Prove that $X_n \xrightarrow{a.s.}0$ as $n \rightarrow \infty$.