Question 6.3: Show that if A ≤T B and B ≤T C, then A ≤T C.

Say that M^{B}_{1} decides A and M^{C}_{2} decides B. Use an oracle TM M_{3}, where M^{C}_{3} decides A. Machine M_{3} simulates M_{1}. Every time M_{1} queries its oracle about some string x, machine M_{3} tests whether x ∈ B and provides the answer to M_{1}.
Because machine M_{3} doesn’t have an oracle for B and cannot perform that test directly, it simulates M_{2} on input x to obtain that information. Machine M_{3} can obtain the answer to M_{2}’s queries directly because these two machines use the same oracle, C.