  Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space
$(\Omega,\mathcal{F},\mathbf{P})$. We show that if
$\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each
$w_n\in\mathbb{R}$, then \[{\mathbf{P}}(\limsup A_n)\geq\limsup_{n\to\infty}
\frac{\displaystyle\big(\sum_{k=1}^n{w_k\mathbf{P}}(A_k)\big)^2}{\displaystyle\sum_{i=1}^n\sum_{j=1}^nw_iw_j{\mathbf{P}}(A_i\cap
A_j)}.\]
