given: $(f_{n})$ are continuous functions from $\mathbb{R}$ to $\mathbb{R}$
my work: Let $A=\left \{ x\in \mathbb{R}|\lim_{n\rightarrow +\infty }f_{n}(x)=L<\infty \right \}$
let $x\in A$ and $B(x,r)$ be given for some $r>0$
$y\in B(x,r)\Rightarrow d(x,y)<r\Rightarrow d(f_{n}(x),f_{n}(y))<\varepsilon _{r}$ (since $f_{n}$ is continuous)
then as $n\rightarrow \infty $, $f_{n}(y)\rightarrow f_{n}(x)\rightarrow L\Rightarrow y\in A$
so $A$ is open then it's a borel subset
Anything wrong ?