I definitely don't think this is a good "first example" of an analytic non-Borel set. It does, however, make more sense when put in comparison with the standard example of a co-analytic, non-Borel set:
Given a binary relation $R$ on $\mathbb{N}$, we can code $R$ by a set $X_R\subseteq\mathbb{N}$: let $\langle \cdot,\cdot\rangle$ be a bijection from $\mathbb{N}^2$ to $\mathbb{N}$ (e.g. the Cantor pairing function), and let $$X_R=\{\langle m, n\rangle: mRn\}.$$ Now for a property $\mathfrak{P}$ of relations, let $S_\mathfrak{P}$ be the family of sets coding relations with that property. E.g. $S_{\mbox{linear}}$ is the family of sets which code linear orders of $\mathbb{N}$, and so forth.
It's not hard to see that $S_{\mbox{linear}}$ is Borel, indeed closed. It turns out that a good way to get Borel sets of high complexity, and non-Borel sets, is to look at sets of the form $S_\mathfrak{P}$ for more complicated properties $\mathfrak{P}$.
One important property of relations is well-foundedness: call a relation $R$ well-founded if there is no infinite sequence $a_0R a_1R a_2R...$. And the associated family of sets is $S_{WF}$.
It's not hard to show that $S_{WF}$ is co-analytic. (Strictly speaking, I've described $S_{WF}$ as a subset of $2^\omega$, not $\mathbb{R}$; but it's easy to switch it over, e.g. via the standard bijection between $2^\omega$ and the middle-thirds Cantor set.) But it turns out that $S_{WF}$ is not Borel!
Why? Well, remember that the Borel sets are arranged in a hierarchy, indexed by countable ordinals (the $\Sigma^0_\alpha$ and $\Pi^0_\alpha$ sets, as $\alpha\in\omega_1$). Now, a well-founded relation has a rank - a countable ordinal that measures how complicated it is. Bigger ranks mean, intuitively, that the relation is "closer to being ill-founded". The point is that Borel sets have a kind of "overspill" property: sets of fixed Borel rank that contain reals coding well-founded relations of too high rank, also contain reals coding ill-founded relations. And this means that the set $S_{WF}$ is not, in fact, Borel.
This is of course not a proof - the details are much more complicated. But this is the idea: there's a connection between the complexity of a Borel set, and its ability to distinguish well-founded relations from ill-founded ones.
This motivates the following general intuition:
If I have a set defined in terms of quantification over infinite sequences, rather than only over natural numbers, then that set probably isn't Borel.
There are of course counterexamples, but this is a good rule of thumb. And the example you mention, of course, is of this kind. It's kind of the opposite of $S_{WF}$: $S_{WF}$ was defined by saying that there is no sequence of a certain form, whereas your set is defined by saying that there is a sequence of a certain form. So a natural thing to do is try to find a Borel isomorphism between your set and the complement of $S_{WF}$.
