Math 310, Homework 4

Due Friday, February 11th

 

  1. Problems II, page 118, Problem 20 (i, ii)
  2. Problems II, page 119, Problem 21 (i)
  3. Use the formal definition of limit of a sequence to show that the sequence an=1/n2 has limit 0 (that is, an is what the book calls a “null” sequence).
  4. Use the (negation of the) formal definition of limit of a sequence to show that the sequence bn=n2 does not have limit equal to 0 (that is, bn is NOT what the book calls a “null” sequence).
  5. Let B be a proper subset of a set A (recall: a proper subset of A is a subset not equal to the entire set A).
    Prove that if there exists a bijection f:A →B, then the set A is infinite.

(Hint: try proof by contradiction and using the definition of cardinality of a set in terms of existence of bijection as discussed in class)

  1. Construct an explicit bijection g from the integers Z to the natural numbers N.