Let $\omega$ denote the set of non-negative integers. Let $A$ denote the set of all sequences of numbers from $[0, 1]$ that converge. We have thus a well defined function $l\colon A \to [0, 1]$, where
$$l((x_i)_{i<\omega}) = \lim_{i\to\infty} x_i.$$
I'd like to extend this function $l$ to the set of all sequences of numbers from $[0, 1]$, we assume the product topology here, such that it will be $\Sigma_2^0$ measurable, i.e. the preimage of an open set will be a $\Sigma_2^0$ set. Equivalently, a Baire 1 function (pointwise limit of continuous functions).
The problem is that I can only show that $A$ is a $\Pi_3^0$ set. If I could show it is $\Pi_2^0$, the extension would be possible.
It is quite possible that $l$ cannot be extended in this way but I don't see that proof either.
Thanks for any suggestions.
EDIT: The function $l$ is $\Sigma_2^0$ measurable on the set $A$. It is the pointwise limit of continuous functions, namely $$l((x_i)_{i<\omega}) = \lim_{n\to\infty} \mathrm{proj}_n((x_i)_{i<\omega}),$$ where $\mathrm{proj}_n((x_i)_{i<\omega}) = x_n$.
There is the notion of Banach limit $\text{Lim}$. This is a linear functional defined on the space of all bounded sequences $B(\mathbb{R})$ which for convergent sequences coincides with lim.