Let A = {: : neN, mN, m >r m (a) Prove that inf A 0. (b) Prove that sup A = 1.

5. Let S-(xe Z xmod 12 8) and T-(xe Z| 4x) Prove that S CT. but TS las-(rez|3reZx-12r+7) and T-(reZ| 3re Z, x= 3s-2) Prove that ScT, but TgS.

4. Show that R¹ and R² are not homeomorphic. How about R¹ and R" for n ≥ 2?

Students are asked to prove the following statement: Theorem: A - (BNC) = (A − B)~ (A − C). One student provides the following proof: Disproof: Note that A (BNC) = An (BC) = An (BCUCC) = (ANB) U (ANC) = (A - B) U (AC) (A - B) n(A - C) ● by set difference law by DeMorgan's law by distributive law by set difference law Is the student's proof correct? If the student is correct, then fill in the needed to • make the proof completely correct • make sure each assertion made is fully justified • make the proof written in such a way that a student in the class could follow the logic and be fully convinced that the theorem is true. If the student is incorrect, then • Identify any errors in the proof above. Explain each error. Your explanation should be written to the student who made the error, and should try to help the student understand why what they wrote is incorrect.