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: 
Solution:

Suppose $x \in A \cup C $. Then by definition, we know that $x \in A $ or $x \in C$. By Hypothesis, we are given that $A \subseteq B$, then we must have that $x \in A \implies x \in B$. Hence

$$ x \in A \; \; or \; \;x \in C \implies x \in B \; \; or \; \; x \in C \implies x \in B \cup C$$

Therefore, we have shown that $A \cup C \subseteq B \cup C $ as desired.
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 $.