math131Ahw02 - Homework#2 Spring 2010 Homework solutions by...

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Homework #2 Math 131A Due: Feb. 16 Spring 2010 Prof. Maruskin Homework solutions by: Liem Tran 4.1-4.4 For each set below that is bounded above, list three upper bounds for the set. Otherwise write “NOT BOUNDED ABOVE” or “NBA.” Repeat for lower bounds. For each set, give its supremum and infimum. (i) \ n =1 ± - 1 n , 1 + 1 n ² Upper bounds: 2 , 3 , 4 Lower bounds: - 3 , - 2 , - 1 Suprema: 2 Infima: - 1 (r) \ n =1 ³ 1 - 1 n , 1 + 1 n ´ Upper bounds: 2 , 3 , 4 Lower bounds: - 2 , - 1 , 0 Suprema: no suprema Infima: no infima (s) µ 1 n : n N and n is prime Upper bounds: 1 2 , 1 , 2 Lower bounds: - 2 , - 1 , 0 Suprema: 1 2 Infima: no infima 4.6 Let S be a nonempty bounded subset of R . (a) Prove that inf S sup S . Hint: This is almost obvious; your proof should be short. If a set is well-ordered, the infima is the least of the set and the suprema is the greatest of the set. Thus inf S < sup S . And if there is just one element of the set then inf S = sup S . Thus, inf S sup S . (b) What can you say about
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math131Ahw02 - Homework#2 Spring 2010 Homework solutions by...

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