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.

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).
