$\sigma$-algebra of well-approximated Borel sets
Solution 1:
Instead of just re-posting my first comment as an answer, I thought it would be better to comment briefly on your original question. (As I said in my second comment, your solution in the edit works fine).
Unfortunately (or rather fortunately) your original approach doesn't work as you intended. Here are three simple observations:
- Every closed set in a metrizable space is a $G_{\delta}$. In particular every point is a $G_{\delta}$.
- Recall the following form of the Baire category theorem: "the intersection of countably many dense $G_{\delta}$s in a complete metric space is again a dense $G_{\delta}$". This immediately implies that a countable dense subset of a perfect Polish space (e.g. $\mathbb{Q}$ in $\mathbb{R}$) is an $F_{\sigma}$ that isn't a $G_{\delta}$.
- It is obvious from the second sentence in 1. that a countable dense set is a countable disjoint union of $G_{\delta}$'s. But as I argued, it won't be a $G_{\delta}$ itself in general.
Finally, let me just note that the detour I suggested in my first comment is not strictly necessary, as by unpacking the proof of the equivalent descriptions you should be able to see what to do in order to save your original direct approach. However, it seems rather obfuscating than clarifying what is going on, so I leave that to you in case you're interested.