Basic compactness

Basic compactness

Okay, so this is my first glance at compactness. I've read the definition,
but I'm strugling with an example (actually the introductory example) from
my textbook. Here's the example: Let $S=(0,2)$ and for each
$n\in\mathbb{N}$ let $A_{n}=(\frac{1}{n},3)$ If $0<x<2$, then by the
Archimedian property, there exists a $p\in\mathbb{N}$ such that
$\frac{1}{n}<x$. Thus $x\in A_{p}$, and
$\mathcal{F}=\{A_{n}:n\in\mathbb{N}\}$ is an open cover for $S$. However,
if $\mathcal{G}=\{A_{1}, \ldots , A_{n_{k}}\}$ is any finite subfamily of
$\mathcal{F}$, and if $m=\mathrm{max}\{n_{1}, \ldots, n_{k}\}$, then
\begin{align*} A_{n_{1}} \cup \cdots \cup A_{n_{k}} = A_{m} =
\Big(\frac{1}{m},3\Big). \end{align*}
It follows that the finite subfamily $\mathcal{G}$ is not an open cover of
(0,2)
As you see, the author "just" assumes that it follows. The way I see it,
is that if $m$ is an index of a finite set, then $\frac{1}{m}$ will never
converge to zero, and therefore, it can not be a cover to $S$ at all.
In my book, the topic "compact set" is before metric spaces. Should I read
and learn about metric spaces before compact sets ?
To me, it's a difficult topic (at least at this first glance), so if you
know any good introductory to supplement with book, please don't hesitate
to share.
Thank you in advance.