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Unformatted text preview: MAT 319/320 Correction of Practice Midterm I Problem 1. Show by induction that for any natural number n greaterorequalslant 1 one has 3 n >n . Proof. 1. The proposition is true for n = 1 because 3 > 1 . 2. Assume that 3 n > n : then 3 n +1 > 3 n = n + 2 n > n + 1 because n greaterorequalslant 1 .Therefore we proved that 3 n +1 >n + 1 . square Problem 2. Determine the set A of all x in R such that | 5 .x + 2 | < 8 . Proof. | 5 .x + 2 | < 8 is equivalent to- 8 < 5 x + 2 < 8 whic is equivalent to- 10 < 5 x < 6 , there- fore the set A is the open interval (- 2 , 6/5) . square Problem 3. Is the set B = braceleftBig 1 n 2 +1 , n N bracerightBig bounded above? bounded below? Does it have a least upper bound, a greatest lower bound? Proof. For any n N one has lessorequalslant 1 n 2 + 1 lessorequalslant 1 therefore it is bounded above and below. Since it is nonempty it has a least upper bound and a greatest lower bound (by completeness of R ).Since 1 is an upper bound and is in the set B...
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- Fall '10