2 - Math 310 - hw 2 solutions Friday, 11 Sept 2009 4.1 h,...

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Math 310 - hw 2 solutions 4.1 h, j, n, p; 4.8, 4.14 a Friday, 11 Sept 2009 4.1 For each set below that is bounded above, list three upper bounds for the set. Otherwise write “Not bounded above” or “NBA”. (h) n =1 [2 n, 2 n + 1] This set is not bounded above (by the Archimedean Property). ± (j) { 1 - 1 3 n : n N } Three upper bounds are 1 , 2 , 3 (note that for n N , we have 1 3 n > 0 and so 1 - 1 3 n < 1). ± (n) { r Q : r 2 < 2 } Three upper bounds are 2 , 3 , 2. ± (p) { 1 , π 3 2 , 10 } Three upper bounds are 10 , 11 , 12 (note that 10 is the maximum of the set). ± 4.8 Let S and T be nonempty subsets of R with the following property: s t for all s S and t T . (a) Observe that S is bounded above and that T is bounded below. Since both S and T are nonempty, there are elements s 0 S and t 0 T . By the given property we have s t 0 for all s S and s 0 t for all t T . Hence, t 0 is an upper bound for S and s 0 is a lower bound for T . It follows that
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This note was uploaded on 03/26/2010 for the course MATHEMARIC 131a taught by Professor Janeday during the Spring '10 term at San Jose State University .

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