I am trying to prove the following identity for a sequence of independent events $A_n$;
$$\mathbb{P}({\limsup_{n\to \infty} A_n})=1-\lim_{n\to\infty}\lim_{N\to\infty} \prod_{m=n}^N(1-\mathbb{P}(A_m))$$ but I am getting stuck: \begin{align}\mathbb{P}(\limsup_{n\to\infty } A_{n})&=\mathbb{P}(\bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m) \\&=\prod_{n=1}^\infty\mathbb{P}(\bigcup_{m=n}^\infty A_m) \\&=\prod_{n=1}^\infty\left(1-\mathbb{P}\left(\left\{\bigcup_{m=n}^\infty A_m\right\}^c \right)\right) \\&=\prod_{n=1}^\infty\left(1-\mathbb{P}\left(\bigcap_{m=n}^\infty A^c_m \right)\right) \\&=\prod_{n=1}^\infty\left(1-\prod_{m=n}^\infty \mathbb{P}\left(A^c_m \right)\right) \\&=\prod_{n=1}^\infty\left(1-\prod_{m=n}^\infty \left(1-\mathbb{P}(A_m)\right)\right) \\&=\lim_{N\to \infty}\prod_{n=1}^N\left(1-\prod_{m=n}^\infty \left(1-\mathbb{P}(A_m)\right)\right) \end{align}
Note that $$ \begin{align*} P(\limsup A_n) &=1-P(\lim\inf A_n^c)\\ &=1-P(\cup_{n=1}^\infty\cap_{k=n}^\infty A_n^c)\\ &=1-\lim_{n\to\infty}P(\cap_{k=n}^\infty A_n^c)\tag{1}\\ &=1-\lim_{n\to\infty}\lim_{N\to\infty}P(\cap_{k=n}^NA_n^c)\tag{2}\\ &=1-\lim_{n\to\infty}\lim_{N\to\infty}\prod_{k=n}^n(1-P(A_n))\tag{3} \end{align*} $$ where in (1) we used measure continuity from below in (2) we used measure continuity from above and in (3) we used independence of the events $A_n$.