formula

$Q_n(A_n)&=&\int_{\mathbf{R}}1_{A_n}\frac{1}{\sqrt{2\pi}}\exp\bigg(-\frac{(x-\mu_n)^2}{2}\bigg){\rm d}x\\ &=&\int_{|x|>M} + \int_{|x|\leq M}\\ &< &\varepsilon + \int_{|x|\leq M}1_{A_n}\frac{1}{\sqrt{2\pi}}\exp\bigg(-\frac{x^2}{2}\bigg)\exp\bigg(\mu_n x-\frac{\mu_n^2}{2}\bigg){\rm d}x\\ &\leq &\varepsilon + \exp(CM)P_n(A_n)\quad (C=\sup_n|\mu_n|, \exp(-\mu_n^2/2)\leq 1)$