If $x_n$ is not Cauchy then an $\varepsilon>0$ can be chosen (fixed in the rest) for which, given any arbitrarily large $N$ there are $p,q \ge n$ for which $p<q$ and $x_q-x_p>\varepsilon.$
Now start with $N=1$ and choose $x_{n_1},\ x_{n_2}$ for which the difference of these is at least $\varepsilon$. Next use some $N'$ beyond either index $n_1,\ n_2$ and pick $N'<n_3<n_4$ for which $x_{n_4}-x_{n_3}>\varepsilon.$ Continue in this way to construct a subsequence.
That this subsequence diverges to $+\infty$ can be shown using the Archimedes principle, which you say can be used, since all the differences are nonnegative and there are infinitely many differences each greater than $\varepsilon$, a fixed positive number.
$\Bbb R$ is a complicated example. Let's instead consider a simple example. Let $S$ be the space $\{7, 8\}$ with the usual distance. (That is, $d(7,7) = d(8,8) = 0, $ and $d(7,8) = d(8,7) = 1$.) I claim that $S$ is a complete space.
We are not going to show this by considering every cauchy sequence one at a time. As you observed, that is impossible.
Let $C$ be a cauchy sequence and the elements of $C$ are $c_0, c_1, \ldots$. Since $C$ is a cauchy sequence, we have, for any positive $\epsilon$, that elements of the sequence that are far enough out will be closer than $\epsilon$. Formally, for any given positive $\epsilon$ there is some integer $N$ so that for any $i,j>N$, we have $d(c_i, c_j) < \epsilon$.
Take, for example, $\epsilon = \frac12$. Then because $C$ is cauchy, we know there must exist some $N$ such that for any $i,j>N$, we have $d(c_i , c_j) < \frac12$, because that is what it means for $C$ to be cauchy; that is the definition.
But if we have $d(c_i , c_j) < \frac12$ then $d(c_i , c_j) = 0$, because the only possible values of $d(\ldots)$ are $0$ and $1$, and it can't be $1$ because $1$ is not less than $\frac12$. So it must be $0$.
So we conclude that if $C$ is cauchy, then for all sufficiently large
$i$, and $j$ we have $d(c_i, c_j) = 0$ and therefore $c_i = c_j$ for all sufficiently large $i$ and $j$. Thus, every cauchy sequence in $S$ is eventually constant. It might have a mixture of $7$s and $8$s at the beginning, but there must come some point (that's $N$) after which all the elements are $7$s or all the elements are $8$s. If this doesn't happen, then $C$ was not a cauchy sequence.
Since $C$ is eventually constant, there is some value, either $7$ or $8$, call it $L$, such that all sufficiently late elements of $c$ are equal to $L$. That is, there is some $N$ such that for any $i>N$, $c_i = L$, where $L$ is either 7, or 8, depending on which $C$ we actually chose.
Is is then trivial to show that $C$ converges to the value $L$. To show this we need to show that for any positive $\epsilon$, the terms of the sequence are eventually within $\epsilon$ of $L$. In this case we can do even better; we know that the terms of $C$ are eventually equal to $L$. $C$ is therefore a convergent sequence.
Therefore, all cauchy sequences in $S$ converge, and $S$ is therefore a complete metric space.
The same argument goes through just as well if we take the more complicated space $\Bbb Z=\{\ldots, -1, 0, 1, 2, 3\ldots\}$ with the usual distance function $d(i,j) = |i-j|$; the proof is exactly the same.
Mathematics is not (usually) about considering a lot of examples. It is about what properties logically imply which other properties. Just as we can prove that every square number is positive without considering each square individually, we can also prove that every cauchy sequence of elements of $\Bbb Z$ must converge, even without considering each cauchy sequence individually.
Best Answer
Assume there are two subsequence $ x_{\phi(n)} $ and $ w_{\psi(n)} $ such that
$$x_{\phi(n)}\to L_1$$ and $$x_{\psi(n)}\to L_2$$
Given $ \epsilon>0$,
we know that $$\phi(n)\ge n ; \psi(n)\ge n$$
then there exists $N_1, N_2,N_3 \ge 0$ such that $$n\ge \max(N_1,N_2,N_3) \implies$$
$$|L_1-L_2|=|L_1-x_{\phi(n)} +x_{\phi(n)}-x_{\psi(n)}+x_{\psi(n)}-L_2|\le \frac \epsilon 3+\frac \epsilon 3 +\frac \epsilon 3$$
We conclude that $$L_1=L_2$$