Lemma for Kolmogorov's extension theorem

Question

I have following question about a step in a lemma used to prove Kolmogorov's extension theorem:

Here the excerpt (from https://wt.iam.uni-bonn.de/bovier/lecture-notes/ "stochastic processes"):

The construction provides only a converving element $a := \lim_{i \to \infty}\pi_{J_k}x_{n_i} \in \cap _{j=1} ^k K_j$.

The question is why should $a$ have a preimage $x \in S^I$? Especially why should there exist a $x \in S^I$ with $a= \pi_{J_k}(x)$?

Answer 1

Define $s_k \in S$ to be $\pi_k(x^\ell)$ for any $\ell \ge k$. Since $\pi_{J_m}(x^n) = x^m$ for any $m \le n$, it follows that $s_k$ is well defined. Now let $x = (s_k)$.

(Or if you want to do a short cut, quote Tychonoff's Theorem.)