Tips for the Final Exam

To prepare for the final exam, you should thoroughly review your class notes and homework assignments. Remember, the final is comprehensive. Do not forget to review logic: tautologies, logical equivalence, truth tables, quantified statements and their negations. They will appear on the exam. Do not forget to review statements of theorems, definitions, examples, and proofs (particularly, proofs of reasonable length). Review homework problems, particularly those which were graded, or whose solutions have been posted on the web. To give you an idea of what to expect, below are a few problems that would be perfect for the exam.

1)  (a) State the Boundedness Theorem.
(b) Prove the Boundedness Theorem.

2)  (a) State the Max-Min Theorem.
(b) Give an example that if the interval is bounded but not closed, the conclusion of the theorem may not hold.

3)  (a) Define a Cauchy sequence.
(b) Prove that a Cauchy sequence is bounded.

4)  (a) Define the limit of a function.
(b) Prove that the limit of a sum is the sum of the limits.
(Of course, the type of the limit would be specified.)

5) Let f be continuous in R and such that f(q)=0 for every rational q. Prove that f(x)=0 for all x.

Etcetera. I hope this gives you an idea what to expect. If you have questions come and see me on Monday, Dec 16, 3 pm - 4 pm.