Every infinite subset of a countable set is countable.

I think some things can be written in a clearer manner. First, I would change

This implies that there is a sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ where $x_{n}\in A$.

For

This implies we can write $A=\{x_n:n\geqslant 1\}$

or

Let $\{x_n:n\geqslant 1\}$ be an enumeration of $A$.

Then, I would say

If $S$ is a subset of the natural numbers, let $\min S$ denote least element of $S$. Define $S_1=\{k:x_k\in E\}$. $S_2=S_1\setminus \{\min S_1\}$ and in general $$S_{n+1}=S_{n}\setminus\{\min S_1,\ldots,\min S_{n}\}$$ Then define $n_k=\mathscr \min S_k$

I guess the idea is clear: consider the set of subscripts such that $x_k\in E$. By the well ordering of the natural numbers, we can extract a sequence $n_k$ such that $$n_1<n_2<\cdots\\E=\{x_{n_k}:k\geqslant 1\}$$

by considering the first subscript with $x_k\in E$ removing this one from the list and looking at the new first subscript (our second in the list) and so on. Some details should be addressed

$(1)$ The set $S_{n}$ is never empty. Reason: Since $E\subseteq A$; $S_1$ is not empty. Moreover, $E$ is by assumption infinite, thus removing one element every time cannot exhaust it.

$(2)$ The construction exhausts the elements of $E$ -- that is, it is a surjection. Reason: Pick $m$ such that $x_m\in E$. We need to find $k$ such that $n_k=m$. Consider the finite set $\{x_1,\dots,x_m\}$. Keeping the order, remove all elements such that $x_i\notin E$. We're left with a finite set, and it must be the case $\{x_{n_1},\dots,x_{n_k}\}$ for some $k$, and $n_k=m$, by definition of the $n_k$.

$(3)$ The construction is an injection. Reason: By construction, $n_k\neq n_j$ if $j\neq k$ for if $j>k$ then $n_k\notin S_{j}$.

Conclusion We obtain an bijection of $E$ with an infinite subset $F$ of $\Bbb N$. Thus $E\simeq F\simeq \Bbb N$, that is $E\simeq \Bbb N$.