title: If A is NP-complete and if there is a reduction from A to B, does it imply that B is also NP-complete?
body: <p>Suppose that A, B, and C are decision problems. Suppose also that A is polynomial-time reducible to B and that B is polynomial-time reducible to C. If both A and C are NP-complete, then does it imply that B is also NP-complete? </p>
  
  <p>I know that, if A is NP-complete and it is polynomial-time reducible to B, then B is NP-hard. However, in order for a problem to be NP-complete, it must meet (1) it's in NP, and (2) it's NP-hard.</p>
  
  <p>I have no idea how to prove the first requirement of NP-complete.</p>