So I'm trying to solve the following problem from Real Mathematical Analysis (Pugh).
Suppose that $(f_n)$ is an equicontinuous sequence in $C^0([a, b], \mathbb{R})$ (the space of continuous real valued functions defined on the closed interval $[a, b]$) and $p\in [a, b]$ is given.
(a). If $(f_n(p))$ is a bounded sequence of real numbers, prove that $(f_n)$ is uniformly bounded. (I've proved this part)
(c). Can $[a, b]$ be replaced with $(a, b)$? $\mathbb{Q}$? $\mathbb{R}$? $\mathbb{N}$?
Sorry if the question is a little bit stupid, but for (c), I'm having trouble figuring out which $[a, b]$ is Pugh refering to? Are we supposed to replace "$C^0([a, b], \mathbb{R})$" with "$C^0((a, b), \mathbb{R})$", etc, or are we supposed to replace "$p\in [a, b]$" with "$p\in \mathbb {Q}\cap [a, b]$", etc? If it is the latter case, isn't $\mathbb{R}\cap [a, b]$ still $[a, b]$?
I know that we have to find, as in the original version of Arzela-Ascoli Theorem, a countable dense subset of $[a, b]$, like $\mathbb {Q}\cap [a, b]$, but for part (c) of the exercise above, Pugh's wording is really confusing.
Any explanation will be appreciated!