Walter Rudin "Principles of Mathematical Analysis" Definition 3.16, Theorem 3.17. I cannot understand. I am reading Walter Rudin's "Principles of Mathematical Analysis".  
There are the following definition and theorem and its proof in this book.  
Rudin didn't prove that $E \neq \emptyset$.  
Why?
Rudin wrote "If $s^* = -\infty$, then $E$ contains only one element" in the following proof.
But, if $E = \emptyset$, then $s^* = -\infty$ and $E$ contains no element.  
So, I think Rudin needs to prove that $E \neq \emptyset$.  
I cannot understand Rudin's proof.  
Definition 3.16: 

Let $\{ s_n \}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_k} \rightarrow x$ for some subsequence $\{s_{n_k}\}$. This set $E$ contains all subsequential limits, plus possibly the numbers $+\infty$, $-\infty$. 
Put $$s^* = \sup E,$$ $$s_* = \inf E.$$

Theorem 3.17:  

Let $\{s_n \}$ be a sequence of real numbers. Let $E$ and $s^*$ have the same meaning as in Definition 3.16. Then $s^*$ has the following two properties: 
(a) $s^* \in E$. 
(b) If $x> s^*$, there is an integer $N$ such that $n \geq N$ implies $s_n < x$. 
Moreover, $s^*$ is the only number with the properties (a) and (b).  
Of course, an analogous result is true for $s_*$. 

Proof:  

(a)
  if $s^* = +\infty$, then $E$ is not bounded above; hence $\{s_n\}$ is not bounded above, and there is a subsequence $\{s_{n_k}\}$ such that $s_{n_k} \to +\infty$.  
If $s^*$ is real, then $E$ is bounded above, and at least one subsequential limit exists, so that (a) follows from Theorems 3.7 and 2.28.  
If $s^* = -\infty$, then $E$ contains only one element, namely $-\infty$, and there is no subsequential limit. Hence, for any real $M$, $s_n > M$ for at most a finite number of values of $n$, so that $s_n \to -\infty$.  
This establishes (a) in all cases. 


Every sequence in $\overline{\mathbb{R}}$ has a convergent subsequence.
If the sequence is bounded, this is trivial by Bolzano's theorem.
Otherwise, the sequence is unbounded. If it is unbounded above, you can find a subsequence that converges to $+ \infty$. If it is unbounded below, you can find a subsequence that converges to $-\infty$. 
