In proving that $\mathcal{F} = \bigcup_{n=1}^\infty \mathcal{F}_n$ is dense, I think Royden and Fitzpatrick are leaving out quite a few details.. Here's my attempt at filling them in. Given $f \in L^p(\Bbb{R})$, the sequence $f_n = f 1_{[-n,n]} \in L^p[-n,n]$ converges to $f$ pointwise on $\Bbb{R}$, which implies $|f_n|^p \to |f|^p$ pointwise on $\Bbb{R}$. Moreover, since $|f_n|^p$ is increasing, the MCT theorems says
$$\int_\Bbb{R} |f|^p = \lim_{n \to \infty} \int_\Bbb{R} |f_n|^p$$
Lemma 7 of the previous section says this happens if and only if $f_n \to f$ in $L^p(\Bbb{R})$. Now, since $f_n \in L^p[-n,n]$ and $S'[-n,n]$ is a dense subset, there exists $s_n \in S'[-n,n]$ for which $\|f_n - s_n\|_p < \frac{1}{n}$. Hence,
$$\|f - s_n \|_p \le \|f - f_n\|_p + \|s_n - f_n\|_p \to 0$$
as $n \to \infty$, because both $\|f - f_n\|_p$ and $\|s_n - f_n\|_p$ go to $0$ as $n \to \infty$. Since $s_n \in \mathcal{F}_n$ for each $n$, we have that there is a sequence of points in the countable set $\mathcal{F}$ converging to $f$, which makes $\mathcal{F}$ dense. Hence $L^p(\Bbb{R})$. Is this likely what Royden-Fitzpatrick had in mind?