Prove A is a subset of B and B is a subset of C if and only

Prove A is a subset of B and B is a subset of C if and only if A union B is a subset of C.

Solution

Let A B and B C. Now, if x AUB, then x A or x B . In either case x C . Therefore AUB C.

While proving the converse, there seems to be an error. If AUB C , then A C and B C ( as x A or x B x C so that A C and B C ). However, AUB C does not imply that A B.