On the Borel-Cantelli Lemma and its Generalization

Let $\{A_n{\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of events on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbf{P})$. We show that if ${\lim }_{m\to \mathrm{\infty }}{\sum }_{n=1}^m{w}_n\mathbf{P}(A_n)=\mathrm{\infty }$ where each $w_n\in \mathbb{R}$, then $$\mathbf{P}(\mathrm{lim\hspace{0.17em}sup}{A}_n)\ge \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{{\displaystyle (\sum _{k=1}^n{w}_k\mathbf{P}(A_k){)}^2}}{{\displaystyle \sum _{i=1}^n\sum _{j=1}^n{w}_i{w}_j\mathbf{P}(A_i\cap A_j)}}\mathrm{.}$$