What does it mean to converge to a point if it is not clear what the point even is? What does it mean for a sequence to converge to an element if the limit might not necessarily be defined or known, or not necessarily in the universe under consideration (whatever this means)?
I am not just talking about real numbers; it can be more general. The definition of a sequence $(x_n)$ converging to $x$ seems to say: for any $\epsilon > 0$, there is $N \in \mathbb{N}$ such that $|x_n - x| < \epsilon$ for all $n > N$. But doesn’t this assume the existence of a point $x$ under consideration? For example, when we show that $\{p \in \mathbb{Q}: p^2 < 2\}$ “approaches” a number, e.g. by considering $1.4, 1.41, 1.414, ...$, what does that even mean, if we have not yet constructed the real numbers? What is meant by “number” in this case? Does it even make sense to say that?
Although we might not "know" what the limiting point $x$ is for $\{p \in \mathbb{Q}: p^2 < 2\}$, it seems to still make sense to speak of a limit. (Again, I'm assuming we do not as yet know what real numbers might be, or what completeness is, etc.). In such a case, how should we define convergence, if we cannot have a point to explicitly refer to?
In general, couldn’t we converge to “something”, but it is not at all clear what that “something” should or might be? If it isn’t clear what that something should be, then how can we even speak of converging to that something? Is this just a logical/semantic/notational thing in the definition of convergence?
 A: Not sure where the confusion comes from but...
In any book of real analysis (e.g. in the simplest case - real analysis of functions of one variable), real numbers are introduced first, and only then the book starts talking about limits. So first the existence of real numbers is shown or at least assumed/postulated via a set of axioms (and rigorously introduced later via Dedekind cuts/Cauchy sequences). And only then limits are discussed.
So yes, the two things must come in the right order and they usually do.
If you stick to rationals but have no real numbers at your side,
there's no real analysis and hence no limits. Right?
Same for analysis of real multivariate functions, complex analysis etc. First you need to know or postulate (via some set of axioms) that "points" exist, and only then talk about sequences converging to points.
Maybe the confusion comes from calculus
(which is basically analysis without too much rigor as far as I know).
On a more philosophical (or funny) level... I've had a few discussions with a physicist who thinks that in a way real analysis is in general somewhat "flawed" but still yields useful results. Why flawed? Because in nature/physics infinitely small positive numbers simply do not exist. E.g. the smallest distance is the Planck length. So how come in analysis we have those $$\epsilon \gt 0$$ values as small as we want them to be?! :)
But of course

*

*As far as I know in math we are allowed to have absolutely abstract theories and concepts which don't map necessarily to natural phenomena.

*I don't know if the Planck length will still be the smallest known distance in several hundred years.

So at least to me this argument which refers to nature is not right here, and I am personally fine with real analysis.
A: This is why the definition of convergence assumes a metric space (or more generally a topological space, but let's stay in metric spaces):
Let $(X,d)$ be a metric space and $(x_n)_{n\in\mathbb N}$ a sequence in $X$. It is said to converge to $x\in X$ if for arbitrarily small $\varepsilon>0$ there exists a sufficiently large $N\in\mathbb N$ such that $d(x,x_n)<\varepsilon$ for all $n\geq N$.
$d(x,y)$ is essentially a generalization of the distance between $x$ and $y$. In the rational or real numbers, it's usually what we would intuitively think of as distance, namely $d(x,y)=\vert y-x\vert$. That's why $\vert x-x_n\vert$ comes up in your less general definition. Anyway, with this more rigorous definition, the answer to your question is: Convergence is very much defined by the underlying set. If there is some bigger set containing $X$ and $x_n$ "converges" to an element $y$ in that bigger set, but which is not in $X$, then we just don't say that $x_n$ converges. With this in mind, we have to make a careful distinction when determining the convergence of the sequence $1,1.4,1.41,1.415,\dots$: If we consider the rationals as the underlying set, then it does not converge, since the "limit" $\sqrt2$ is not in the underlying set. But if we consider the reals as the underlying set, it does converge.
There is also a certain notion of when a sequence "should" converge. If the members of the sequence get arbitrarily close to each other for large enough index $n$, then we would intuitively expect it to converge. Such a sequence is then called Cauchy sequence, named after French mathematician Augustin-Louis Cauchy, who just assumed it as obvious that a sequence converges if it is a "Cauchy sequence". But it turns out, we have to be more careful, because not every Cauchy sequence converges. At least not in every metric space, because the point to which such a sequence should converge might not be in the underlying set. Such a metric space is called incomplete, otherwise it is called complete. But for every incomplete metric space, there is a bigger, complete metric space in which every Cauchy sequence does converge. Such a metric space is then called the completion of the smaller space. For instance, the real numbers are the completion of the rational numbers: There are rational sequences converging to an irrational number because the rationals are incomplete, and there is no larger space than the reals containing additional points to which a real sequence might converge, because the reals are complete.
