Set 1

(there are a total of 9 problems):

  1. (Rudin Ch. 2) #12
  2. (Rudin Ch. 2) #16
  3. (Rudin Ch. 2) #17
  4. (Rudin Ch. 2) #26
  5. (Bergman (Links to an external site.)) #2.3:1;
  6. (Bergman (Links to an external site.)) #2,3:3 (assume K is infinite)
  7. (Problems (Links to an external site.)) X4- either verify that D is a metric or complete the problem as written;
  8. Let A denote a non-empty bounded subset of the real line. Let s denote the supremum of A.  Let e denote a positive number.  Use the definition of supremum to prove that there exists a point a in A so that s-e < a.
  9. Find a bijection between the real numbers and the subset of the real line, (0,1).  Prove that your map is injective and surjective.