The author of the book I'm reading wants me to use the following lemma:
There exists an open set in $\mathbb{R}^2$ such that every open set in $\mathbb{R}$ is some vertical section in it, there exists a $G_\delta$ set in $\mathbb{R}^2$ such that every $G_\delta$ set in $\mathbb{R}$ is some vertical section in it, same with $G_{\delta\sigma}$, $G_{\delta\sigma\delta}$, etc.
Suppose $G_\delta=F_\sigma$ for a contradiction and let $A\subseteq\Bbb R^2$ be a $G_\delta$ set such that every $G_\delta$ subset of $\Bbb R$ appears as a section of $A$. Let $$B=\{x\in\Bbb R\mid (x,x)\not\in A\}.$$
Using $G_\delta=F_\sigma$ it's easy to check that $B$ is $G_\delta$, so it must be equal to the $y$ section of $A$ for some $y\in\Bbb R$. Trying to compute wether $y\in B$ will lead to a contradiction.
The argument for $G_{\delta\sigma}$ and higher levels is exactly the same.