Prove if A is a subset of B, and B is a subset of C, then A is a subset of C.

Prove if A is a subset of B, and B is a subset of C, then A is a subset of C.

Assume A is a subset of B, and B is a subset of C.

Then there is an x in B such that, for every x in B x is in C and there exists an x in A such that, for every x in A x is in B. Thus, for every x in A, x is also in C, and therefore A is a subset of C.