Assignment 2, Math 220
Due: Friday, January 25th, 2002
1 Let
S = {1 − (−1)n /n : n ∈ N} .
Find inf S and sup S and prove your answers.
2 Let S ⊆ R and suppose that s∗ = sup S belongs to S. If u 6∈ S,
show that
sup (S ∪ {u}) = sup{s∗ , u}.
3 Show that a nonempty finite set S ⊆ R contains its supremum.
Hint: Use Mathematical Induction and the previous question.
4 Question 12.4, page 113 of Lay (4th edition).
5 Suppose that a and b are rational numbers with a < b. Give an
explicit example of an irrational number α with a < α < b.
6 Question 13.4, page 121 of Lay (4th edition).
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