This is an exercise from "Baby Rudin" and I don't understand one step of a particular proof based on the textbook's hint.
$\mathbf{Statement:}$Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.
$\mathbf{Hint:}$ By Exercises 23 and 24, X has a countable base. It follows that every open cover of X has a countable subcover$G_n$, n=1,2,3,… If no finite subcollection of $G_n$ covers X, then complement $F_n$ of
$G_1 \cup.. \cup G_n$ is nonempty for each n, but $\cap F_n$ is empty. If E is a set which contains a point from each $F_n$, consider a limit point of E, and obtain a contradiction.
$\mathbf{Proof:}$ Following the hint, we consider a set consisting of one point from the complement of each finite union, e.t.,$x_n\notin G_1∪⋯∪G_n$. Then E cannot be finite. By hypothesis E must have a limit point z, which must belong to some set $G_n$；and since $G_n$ is open, there is a $\epsilon >0 $ such that $B(z,\epsilon) \subset G_n$ . But then $B(z,\epsilon)$ cannot contain $x_m$ if $m \gt n$, and so z cannot be a limit point of ${x_m}$. We have now reached a contradiction.
$\mathbf{Question:}$ I don't understand the causal relation between "z cannot be a limit point of ${x_m}$." and that z is not a limit point of E.
Best Answer
IIRC Rudin defines a limit point of $E$ as a point $x$ such that $B(x,r) \cap (E \setminus \{x\}) \neq \emptyset$ for every $r>0$. He then has a lemma of sorts that shows that :
If $x$ is a limit point of $E$, then for each $r>0$, $B(x,r) \cap E$ is infinite. This uses the fact that all finite subsets of a metric space are closed.
Then to the proof: suppose that $G_n, n \in \mathbb{N}$ is a countable open cover without a finite subcover. The no finite subcover condition allows us to take a point $x_1 \notin G_1$ ,$x_2 \notin G_1 \cup G_2$, and in general $x_n \notin \cup_{i=1}^n G_i$. Then define $E = \{x_n: n \in \mathbb{N}\}$ which is an infinite set. By assumption $E$ has a limit point $p$. This $p$ is covered, so for some $m$, $ \in G_m$, and as $G_m$ is open, for some $r>0$, $B(p,r) \subset G_m$. Now, for all $n>m$, by construction $x_n \notin \cup_{i=1}^n G_i$, so in particular $x_n \notin G_m$(as $n > m$), so $x_n \notin B(p,r)$.
So $B(p,r) \cap E \subset \{x_n: n \le m\}$, as all $x_n$ with larger index have been ruled out above. So $B(p,r) \cap E$ is finite. So by the first paragraph's equivalence, $p$ cannot be a limit point of $E$. Contradiction.
So $X$ is countably compact: every countable open cover has a finite subcover. As you already know that $X$ has a countable base we can thin out any open cover of $X$ to a countable one first, and a finite one now, so compactness has been shown.
It's important to work with the infinite intersection variant of limit points here.