Suppose that A,B and C are sets such that A is the improper subset of B and B is the improper subset of C

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An improper subset is a subset containing every element of the original set.

Transitive Property of Subsets: For all sets $A, B,$ and $C:$

if $A ⊆ B$ and $B ⊆ C,$ then $A ⊆ C,$

if $C ⊆ B$ and $B ⊆ A,$ then $C ⊆ A.$

Therefore we prove that if $A$ is the improper subset of $B$ and $B$ is the improper subset of $C,$ then $A$ is the improper subset of $C.$