Proof Verification: If $A \subset B$ and $B \subset C$, then $ A \cup B \subset C$ I am trying to prove that:

If $A \subset B$ and $B \subset C$, then $ A \cup B \subset C$

My proof is :
Given some $x \in A \cup B$, it is true that either $x \in A$ and/or $x \in B$. IN the case that $x \in A$  it is true that $x \in B$, as $A \subset B$, and that $x \in C$ , as $B \subset C$. In the case that $x \in B$ it is true that $x \in C$, as $B \subset C$. Therefore, $A\cup B \subset C$
Is this correct?. Any tips to improve this would be appreciated as I am self taught and new to proof writing.
 A: Just to answer  : yes, the approach is correct.
You could also prove that $A∪B=B$, and then the fact follows from $B=A∪B\subset C$. 
For example, if $x \in B$ is true, then of course $x$ is in $A$ or $x$ is in $B$ is true, so $B \subset A \cup B$. If $x \in A$, then $x \in B$ because $A \subset B$, and if $x \in B$, then of course $x \in B$, so $A \cup B \subset B$, hence $A \cup B = B$.
A: Another approach is by using Venn diagrams. Draw circles $A$, $B$ and $C$ for three sets such that $A$ is contained in $B$ and $B$ is contained in $C$ (according to given set inclusions). So you have $A$ as the innermost, $B$ in the middle and $C$ as the outermost of them. Now $A\cup B$ is given by the middle circle which is offcourse contained in $C$ (the outer circle).
A: The most natural way seems to be:


*

*(Transitivity) From $A \subset B$ and $B \subset C$, show $A \subset C$.


If $x \in A$, then $x \in B$.
If $x \in B$, then $x \in C$.
Thus if $x \in A$, then $x \in C$.


*From $A \subset C$ and $B \subset C$, show $(A \cup B) \subset C$.


If $x \in (A \cup B)$, then $x \in A$ or $x \in B$.
But if $x \in A$, then $x \in C$, and if $x \in B$, then $x \in C$.
Thus, either way, $x \in C$.
You implicitly used this method in your (correct) proof, but you didn't separate out the ideas. It can be especially useful to organise larger proofs in terms of simpler definitions, lemmas and theorems.
A: 
Is this correct?

I believe that your proof regarding this Question is correct.

Any tips to improve this would be appreciated as I
  am self-taught and new to proof writing.


Set Theory which you are using (Elementary-Set Theory) was mostly Contributed for its Development by George Cantor.
But, in Later Stage, there are some famous Loopholes found in this Theory, the most popular of them is Russel's Paradox. After Russel's paradox, numerous other Paradox came after which ZFC Set Theory (Zermelo Franklin Set Theory) came which is based on some of the Basic Set Axioms (Related to Mathematical Logic)

Proofs in Set Theory
Most of the Proofs in Elementary Set Theory is based on Mathematical Logic. 
Some of the Basic Theorems and Properties are below -


*

*De Morgan's Theorem -https://en.wikipedia.org/wiki/De_Morgan%27s_laws

*Complement - https://en.wikipedia.org/wiki/Complement_(set_theory) $ A^` = U - A$

*Intersection and Union - 



Alternate Methods to Proof Set Theory Question


*

*Using Venn Diagram - https://en.wikipedia.org/wiki/Venn_diagram
Your Question Can be Proved Using Venn Diagram too like here - 



*

*Using The Laws and Properties of Sets (Intersection, Complement and Union)

