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# assignment1 - lim sup n →∞ x n y n< lim sup n →∞...

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MATH 255 ASSIGNMENT 1 This assignment is due in class on Monday, January 16 Problems Please justify carefully your answers. 1. [10 points] Let ( x n ) be a bounded sequence of real numbers. Prove that lim inf n →∞ x n = sup { t : { n : x n < t } is ﬁnite } . 2. [10 points] Let ( x n ) be a sequence of real numbers. Prove that lim sup n →∞ ( - x n ) = - lim inf n →∞ x n . 3. [20 points] Let ( x n ) and ( y n ) be bounded sequences of real numbers. (1) Prove that lim sup n →∞ ( x n + y n ) lim sup n →∞ x n + lim sup n →∞ y n . (2) Find and example of ( x n ) and ( y n ) such that
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Unformatted text preview: lim sup n →∞ ( x n + y n ) < lim sup n →∞ x n + lim sup n →∞ y n . (3) Suppose that lim n →∞ y n exists. Prove that lim sup n →∞ ( x n + y n ) = lim sup n →∞ x n + lim n →∞ y n . 4. [10 points] Find lim sup n →∞ x n and lim inf n →∞ x n in the following cases: (1) x n = sin( nπ/ 4). (2) x n = ± 1 + 1 n ² n cos( nπ ) . 5. [10 points] Let x n = sin( n ). Find lim sup n →∞ x n . 1...
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