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 $x \in B \cup C$.
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.