Prob. 57, Chap. 1, in Royden's REAL ANALYSIS: The set of points at which a sequence of continuous functions converges is a ... Here is Prob. 57, Chap. 1, in the book Real Analysis by H.L. Royden and P.M. Fitzpatrick, 4th edition:

Let $\left\{ f_n \right\}$ be a sequence of continuous real-valued functions defined on $\mathbb{R}$. Show that the set of points $x$ at which the sequence $\left\{ f_n(x) \right\}$ converges to a real number is the intersection of a countable collection of $F_\sigma$-sets.

And, an $F_\sigma$-set is a set that can be expressed as the union of countably many closed sets.
My Attempt:

Let $S$ be the set of all the real numbers $x$ for which the sequence $\left\{ f_n(x) \right\}$ converges to some real number. Let us now define the function $f \colon S \longrightarrow \mathbb{R}$ by the formula
$$
f(x) := \lim_{n \to \infty} f_n(x).
$$

Is the way I've started out likely to reach the desired conclusion? If so, then how to proceed from here?
 A: With problems like this, it is usually more helpful to express the definition of convergence in terms of unions and intersections rather than defining a function on your set. I'll include the full solution below so be warned.
Let $S$ denote the set of points $x$ where $\lim_{n \to \infty} f(x)$ exists. Since a sequence of real numbers is convergent if and only if it is cauchy, we will be using the cauchy condition. As such, $x \in S$ if and only if for all $\epsilon > 0$ there is some $N \in \mathbb{N}$ with $|f_n(x) - f_m(x)| < \epsilon$ for all $n,m \geq N$. We can then restate this as $x \in S$ if and only if for all $M \in \mathbb{N}$ there is some $N$ with $|f_n(x) - f_m(x)| \leq 1/M$ for all $n,m \geq N$. Translating this into unions and intersections we get that $$S = \bigcap_{M \in \mathbb{N}} \bigcup_{N \in \mathbb{N}}\bigcap_{m,n \geq N} |f_n - f_m|^{-1}([0, 1/M]).$$ Since $|f_n - f_m|$ is continuous and $[0,1/M]$ is closed, it follows that $\bigcap_{m,n \geq N} |f_n - f_m|^{-1}([0, 1/M])$ is closed. Hence, $S$ is an intersection of $F_{\sigma}$ sets.
