I am working on the following exercise from Royden's Real Analysis (Chapter 10, Section 10.1 on the Arzela-Ascoli Theoerem):
Let $S$ be a countable set, and $\{ f_{n} \}$ a sequence of real-valued functions on $S$ that is pointwise bounded on $S$. Show that there is a subsequence of $\{ f_{n} \}$ that converges pointwise on $S$ to a real-valued function.
So far, this is what I have:
Suppose $\{f_{n}\}$ is a sequence of real-valued functions defined on a countable set $S$ such that $\forall x \in S$, $|f_{n}(x)|\leq M(x)$.
Since $f_{n}(x) \in \mathbb{R}$ $\forall x \in S$, this sequence is bounded in $\mathbb{R}$. Therefore, by Bolzano-Weierstrass, it possesses a convergent subsequence $f_{n_{k}} \in \mathbb{R}$.
However, this just seems too easy, and I never used the fact that $S$ was countable anywhere, so I don't think that this is right.
I'm wondering if I should use the Arzela-Ascoli Lemma:
Let $X$ be a separable metric space and $\{f_{n}\}$ an equicontinuous sequence in $C(X)$ (the linear space of continuous real-valued functions on $X$) that is pointwise bounded. Then a subsequence of $\{f_{n}\}$ converges pointwise on all of $X$ to a real-valued function $f$ on $X$.
But, I'm not told that $S$ is separable, or even a metric space (it's countable. Can I say that it's dense in itself, and thus be able to say that it's separable?). Also, I have to admit I'm not 100% on showing that something is equicontinuous.
Here's the definition of equicontinuity:
A collection $\mathcal{F}$ of real-valued functions on a metric space $X$ is said to be equicontinuous at the point $x \in X$ provided for each $\epsilon > 0$, $\exists \delta > 0$ such that $\forall f \in \mathcal{F}$ and $x^{\prime} \in X$, if $\rho(x^{\prime},x)<\delta$, then $|f(x^{\prime})-f(x)|<\epsilon$.
But, how can I show that $\{f_{n}\}$ is equicontinuous? Is this where the countability comes into play?
Please help!