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midtersolnmat25 - Midterm Solutions 1 Denitions State the...

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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 0 for all s 0 S . s is the least upper bound of S , which means whenever there’s some other upper bound t of S , s t . (b) (3 pts) The sequence of real numbers ( x n ) converges to x R . Solution: ε > 0, N so that n > N , | x n - x | < ε . (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 ε > 0, M so that N > M , | v N - a | < ε . 1
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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 non-empty 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 it’s 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 1 - 1 k for some k .
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