2. Rules of the game  You will work with your Unit 3 group  We will have 4 rounds of 6 questions each. Most are multiple choice; some are fill in the blank.  Point values per question:  Rounds 1 and 2: 1pt;  Rounds 3 and 4: 2pts.  You will be required to write down one answer per round (i.e., to choose one answer you are particularly confident about)  If you get it right, you get points for that question  If you get it wrong, you lose points! We will play this game on the HONOR system with each team keeping track of its own points. In the end, the team with the highest score will win a prize!

3. Each question is worth 1 point.

4. Suppose f : R → R is continuous and define the zero set of f by Z(f)= { x: f(x) = 0} How would you prove that Z(f) is a closed set?

6. Suppose f,g: R → R are continuous functions such that f(r) = g(r) for all r in Q. That is to say, f and g are equal on the rational numbers. Prove that f(x) = g(x) for all x in R.

7. Find a continuous function f:(0,1) → R and a Cauchy sequence (x n ) such that f(x n ) is not a Cauchy sequence. Find a continuous function f:[0,1] → R and a Cauchy sequence (x n ) such that f(x n ) is not a Cauchy sequence.

8. Let A denote a closed subset of R. Prove that for all B ⊆ A, if B has a supremum, then sup(B) ∈ A.

9. Let (a n ) be a sequence real numbers which satisfies the property that |a n+1 − a n | ≤ ½ |a n −a n−1 | for all n>1. Prove that (a n ) is a Cauchy sequence?

10. Let (a n ) be a sequence real numbers and assume that (a 2n ) and (a 2n+1 ) converge to the same limit. Does (a n ) converge? Prove your answer.

11. Let E=(0,1]. For n≥1, let O n =(0,1+1/n). Is {O n :n≥1} an open cover of E? Is E compact?

12.  Give an example of a nonempty finite set which is neither open nor closed?

13. Prove that [0,1) is not open. Prove that [0,1) is not closed.

14. Find an open cover of Q with no finite subcover. Find a open cover of Q with a finite subcover.

15. Find an example of two monotone sequences (a n ) and (b n ) where their sum (a n + b n ) is not monotome. Prove that if both (a n ) and (b n ) are increasing, then (a n + b n ) is increasing.

16. Let A and B be two countably infinite sets. Prove that there is a bijection f:A  B.

17.  Prove that the set of all ordered pairs of rational numbers is countable.  Prove that the natural numbers N can be expressed as a countable union of disjoint countably infinite sets.

18. Prove that if (x n ) is a sequence of positive real numbers that diverges to infinity, then the sequence (1/ x n ) converges to 0.

19. Let (x n ) be a sequence of positive real numbers that converges to L. Let p be a fixed positive integer. Prove that the sequence (x n+p ) also converges to L.

20. Prove that |a| - |b| ≤ |a-b|

21. Prove that any interval (a,b) in the real numbers contains an uncountable number of real numbers. [ You can use the fact that (0,1) is uncountable].

22.  Find 1/5 in base 3.

23. How far apart are the points a=(1,3) and b = (4,7) In the taxicab metric? In the max metric? In the Washington DC metric?

24. Find a continuous function f: R → R and an open set U such that F(U) is not open. Find a continuous function f: R → R and a closed set C with f(C) not closed?

25. Give a sequence of functions f n :[0,1]  R that is pointwise convergent but not uniformly convergent.