Question: Good night, someone could help me with some interesting applications of this important theorem and also I could say how I draw a dense subset in the complete space used in the theorem.
Thanks for your help

Answer: 
Your proof is not quite correct. The idea of element chasing is as follows: 

To show that $A\cup C \subseteq B \cup C$: You want to show that any element of $A \cup C$ is also an element of $B \cup C$. 

To that end you would say, suppose $x \in A \cup C$. 

(You fill in the blank here!) 

and conclude that $x \in B \cup C$.
Suppose $x \in A \cup C $. Then $x\in A$ or $x \in C$. 

Case 1. Assume $x \in C$. So $x \in B \cup C$. 

Case 2. Assume $x \in A$. Then because $A \subseteq B$, $x \in B $. Thus $x\in B \cup C$. 

So if $x \in A \cup C $ and $A \subseteq B$, then $ x \in B \cup C $.