Let $\mathbb{R}^{\mathbb{N}}$, the space of real-valued sequences with it natural topology and the corresponding Borel $\sigma $-algebra. I want to see if the subset $S:=\{x\in \mathbb{R}^{\mathbb{N}}: \sum_{n\geqslant 1}x_n\in \mathbb{R}\}$ is Borel. As far as I can see, using the metric
$$ d(x,y):=\sum_{n\geqslant 1}\frac{|x_n-y_n|}{2^n (1+|x_n-y_n|)} $$
the set $S$ is neither open nor closed. But I don't have a clue about how I can continue from there. Can someone give me an answer or a reference related to this question?
It seems that the statement is true. To see this observe that the map $\{x_n\}_{n\in\mathbb{N}}\mapsto \{x_n\}_{n\leqslant m}$ is trivially measurable for every $m\in \mathbb{N}$ as the topology in $\mathbb{R}^{\mathbb{N}}$ is the product topology and so in this case the Borel $\sigma $-algebra coincides with the $\sigma $-algebra generated by the cylinders (because $\mathbb{N}$ is countable and $\mathbb{R}$ is second countable).
So let $f_m:\mathbb{R}^{\mathbb{N}}\to \mathbb{R}^m,\, x\mapsto (x_1,\ldots ,x_m)$, now as $\mathbb{R}^m$ is a topological vector space we have that the map $s_m:\mathbb{R}^m \to \mathbb{R},\, (x_1,\ldots ,x_m)\mapsto \sum_{k=1}^m x_k$ is continuous, therefore the functions defined by $h_m:=s_m\circ f_m$ are measurable for each $m \in \mathbb{N}$. Thus, from basic results of measure theory, we knows that the functions defined by $$ ls:\mathbb{R}^{\mathbb{N}}\to \overline{\mathbb{R}},\, x\mapsto \limsup_{m\to\infty}h_m(x)\\ li:\mathbb{R}^{\mathbb{N}}\to \overline{\mathbb{R}},\, x\mapsto \liminf_{m\to\infty}h_m(x) $$
are also measurable. Now the set in question is just the set defined by $$ \left\{x\in \mathbb{R}^{\mathbb{N}}: ls(x)=li(x) \,\land\, ls(x)\in \mathbb{R}\right\} $$ and so it is measurable.∎