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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). 1