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Unformatted text preview: Midterm Solutions 1. Definitions. State the formal definitions of the following: (a) (3 pts) s is a supremum of the set S R . Note: make sure you define terms such as upper bound. Solution: s must satisfy two conditions: s is an upper bound of S , which means s s for all s S . s is the least upper bound of S , which means whenever theres some other upper bound t of S , s t . (b) (3 pts) The sequence of real numbers ( x n ) converges to x R . Solution: > 0, N so that n > N ,  x n x  < . (c) (3 pts) The lim sup of a sequence ( s n ) is the number a R . Solution: Let v N = sup { s n : n > N } . Then a = lim N v N . This means > 0, M so that N > M ,  v N a  < . 1 2. Examples. Give examples of the following: (a) (2 pts) A logical statement which is a contradiction. Solution: p p , or p p . (b) (3 pts) A nonempty subset A of R that is bounded above but does not have a maximum. Solution: A = 1 1 n : n 1 A is bounded above by 1 (which is its supremum). But, A has no largest element. To verify this, suppose by contradiction that A had a largest element. Then it would be of the form 1element....
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.
 Winter '11
 Wiley
 Differential Equations, Equations

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