Prove the set for which a sequence of continuous functions converge is Borel

Question

I've been trying to solve the following problem:

Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be a sequence of continuous functions. Prove that the set of points $E = \{x \in\mathbb{R}\mid\{f_n(x) \text{ converges }\}\}$ is Borel.

Note that $E$ is precisely the set of points for which $\{f_n(x)\}$ is Cauchy. So my idea is, for $x \in E$, to start taking neighborhoods around $x$ with radius $\delta_1, \delta_2, \ldots$ such that the elements in a neighborhood with radius $\delta_i$ are "close enough" $f_i(x)$, but I am having troubles to formalize this idea.

Answer 1

$E=\cap_n \cup_m \cap_{j,k \geq m} \{x:|f_j(x)-f_k(x)| <\frac 1 n\}$. The sets $\{x:|f_j(x)-f_k(x)| <\frac 1 n\}$ are open.