Solution
= Solution
Write $p_n=\mathbb P(A_n)$. Independence gives, first for finite partial counts and then by monotone convergence,
$$
\mathbb E[2^{-N}]
=\prod_{n\geq1}
\mathbb E[2^{-\mathbf1_{A_n}}]
=\prod_{n\geq1}\left(1-\frac{p_n}{2}\right).
$$
Using $1-x\leq e^{-x}$ factor by factor,
$$
\boxed{
\mathbb E[2^{-N}]
\leq\exp\left(-\frac12\sum_{n\geq1}p_n\right)
=e^{-\mathbb E[N]/2}}.
$$