Consider a probability space \((\Omega, \mathcal{F}, P)\) where \(\Omega=(0,1), \mathcal{F}=\mathcal{B}\{(0,1)\}\) and \(P\) is Lebesgue measure on \((0,1)\). Let \(\left\{A_{n}ight\}_{n=1}^{\infty}\) be a sequence of events in \(\mathcal{F}\) defined by \(A_{n}=\left(0, \frac{1}{2}\left(1+n^{-1}ight)ight)\) for all \(n \in \mathbb{N}\). Show that

\[\lim _{n ightarrow \infty} P\left(A_{n}ight)=P\left(\lim _{n ightarrow \infty} A_{n}ight)\]