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.

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.
