# hw2 - 6. If a n ≤ x n ≤ b n and lim n →∞ a n = lim...

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Homework 2 – Math CS 120, Winter 2009 Due on Thursday, January 22nd, 2009 1. Prove that a convergent sequence cannot converge to two diﬀerent limits. 2. Consider the sequence of real numbers a n = n + 1 - n. Prove that it converges (and ﬁnd the limit), using the deﬁnition. 3. Do the same for a n = n 2 - 1 3 n 2 + 4 n - 1 . 4. Consider a set of real numbers { a i } p i =0 R , such that a 0 + a 1 + a 2 + ··· + a p = 0 . Prove that lim n →∞ ( a 0 n + a 1 n + 1 + a 2 n + 2 + ··· + a p n + p ) = 0 . 5. Prove that if a n > 0 n N , and lim n →∞ a n +1 a n = L, then lim n →∞ n a n = L.

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Unformatted text preview: 6. If a n ≤ x n ≤ b n and lim n →∞ a n = lim n →∞ b n = L , show that lim n →∞ x n = L . (This is sometimes called the sandwich theorem). 7. If b ≤ x n ≤ c for all but a ﬁnite number of n , show that b ≤ lim inf n →∞ x n , and lim sup n →∞ x n ≤ c. 8. Find lim inf n →∞ and lim sup n →∞ when: 1 2 (a) a n = (-1) n ± 2 + 3 n ² . (b) a n = n + (-1) n (2 n + 1) n ....
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hw2 - 6. If a n ≤ x n ≤ b n and lim n →∞ a n = lim...

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