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.