Points of convergence I can't seem to solve the following : 
Let $\left\{f_n(x)\right\}$ be a sequence of nonnegative continuous functions. 
Prove that : 
S = $\left\{x \in R |\sum\limits^{\infty}_{n=1}f_n(x) < \infty \in \right\}$
is a Borel Set. 
I am thinking something along the lines of :
Let $x \in S$, the above sum is finite and therefore converges. So we should prove that  for $x$ there exists an $\epsilon > 0$ such that the sum also converges for  all $y < x + \epsilon$. But how? 
Or we can say since the sequence converges therefore it is Cauchy therefore its points get closer and closer to each other but then how does their inverse make an open interval around $x$? 
 A: Let $N \in \mathbb{N}$. For all $m \in \mathbb{N}$, the function $g_m = \sum_{i=1}^m f_i$ is continuous. Further, $\sum_{n=1}^\infty f_n \leq N$ if and only if $g_m \leq N$ for all $m \in \mathbb N$. Thus we have the set equality $\{x \in \mathbb{R} : \sum_{n=1}^\infty f_n(x) \leq N \} = \bigcap \limits_{m=1}^{\infty} g_m^{-1}([0,N])$. Since $g_m$ is continuous, and $[0,N]$ is closed, this set is closed. Since $S = \bigcup \limits_{N=1}^{\infty} \{x \in \mathbb{R} : \sum_{n=1}^{\infty} \leq N \}$, $S$ is $F_\sigma$.
A: Check that
$$
S = \cup_{k=1}^{\infty} \{x\in \mathbb{R} : \sum_{n=1}^{\infty} f_n(x) \leq k\}
$$
and
$$
\sum_{n=1}^{\infty} f_n(x) \leq k \Leftrightarrow \sum_{n=1}^m f_n(x) \leq k \quad \forall m \in \mathbb{N}
$$
since the $f_n$'s are non-negative. Let $g_m(x) = \sum_{n=1}^m f_n(x)$, then $g_m$ is continuous, and
$$
S = \bigcup_{k\in \mathbb{N}} \bigcap_{m \in \mathbb{N}} g_m^{-1}(-\infty, k]
$$
Note: This is a good exercise in trying to prove a set is Borel. What you usually try to do is to express the set as a sequence of unions/intersections, until you hit upon an open/closed set.
