Prove That A ⊆ B, B ⊆ C and C ⊆ A iff A = C

Solution

Given \[ A \subseteq B,\ B \subseteq C \] therefore \[ A \subseteq C \]

Also, \[ C \subseteq A \]

Hence, \[ A \subseteq C \text{ and } C \subseteq A \] Therefore, \[ A=C \]

Conversely, if \[ A=C \] then clearly \[ A \subseteq C \text{ and } C \subseteq A \]

Hence, \[ \boxed{A \subseteq B,\ B \subseteq C \text{ and } C \subseteq A \iff A=C} \]