Theorem 3.17 in Rudin's Analysis

This question refers to Rudin's "Principles of Mathematical Analysis", Theorem 3.17, p.56. In particular let $\left\{s_n\right\}$ be a real sequence and let $E$ be the set of all sub-sequential limits with possibly plus and minus infinity included. Denote $s^*=\sup E$. Suppose $x$ is a real number such that $x>s^*$ and that $s_n \ge x$ for infinitely many values of $n$. I want to arrive to a contradiction, but i am having some difficulty doing this very rigorously. More precisely, i can see that we obtain subsequences, which after some index become larger than $x$, but why does that mean that we have a sub-sequential limit greater than $x$ (thus contradicting $s^*$)?

Thanks.

It relies on the following fact (prove it as an exercise) :

$I \subset \mathbb{N}$ is infinite if and only if there exists an increasing function $\varphi : \mathbb{N} \to I$ which is bijective.

If you put $I$ the set of all integers $n$ such that $s_n \ge x$, if $I$ was infinite, there would exist, by the previous fact, a subsequence $(s_{\varphi(n)})$ whose all terms are $ \ge x$.

Now, extract a converging sub-subsequence from $(s_{\varphi(n)})$. What can be said about its limit ?

(recall that every sequence of real numbers admits a converging subsequence, possibly in the extended real line)