Q.If $A$, $B$, $C$ are subsets of a universal set $U$ such that $A \subseteq B$ and $B \subseteq C$, prove that $A \subseteq C$. Verify this using $U = \{1, \dots, 10\}$, $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$, $C = \{1, 2, 3, 4, 5, 6\}$.

[!TLDR] Subset containment is transitive: $A\subseteq B\subseteq C \Rightarrow A\subseteq C$. [!ANSWER] $A \subseteq C$ is proved t …

Proof. Let $x$ be any element of $A$, i.e. $x \in A$. Since $A \subseteq B$, it follows that $x \in B$. Since $B \subseteq C$, it then follows that $x \in C$. So every element of $A$ is also an element of $C$, which is exactly the definition of $A \subseteq C$. $\blacksquare$ Verification. With $A=\{1,2\}$, $B=\{1,2,3,4\}$, $C=\{1,2,3,4,5,6\}$: both elements of $A$ (namely 1, 2) are indeed eleme …

Use the definition of subset directly on an arbitrary element $x \in A$, chaining $A\subseteq B$ and $B\subseteq C$ to concl …

A common mistake is trying to "prove" this only by checking one example, without giving the general …