1. Some Review Problems for Math 141 Final

2. 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?

4. 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.

5. Find a continuous function f:(0,1)→R and a Cauchy sequence (xn) such that f(xn) is not a Cauchy sequence.

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

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

8. Let (an ) be a sequence real numbers and assume that (a2n ) and (a2n+1 ) converge to the same limit. Does (an ) converge? Prove your answer.

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

10. Give an example of a nonempty finite set which is neither open nor closed?
(Or explain why no such example can exist)

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

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

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

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

15. 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.

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

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

18. Prove that |a| - |b| ≤ |a-b|

19. 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].

20. 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?

21. 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?

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