sheet1 - CM221A ANALYSIS I Exercise Sheet 1 1. Prove that,...

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CM221A ANALYSIS I Exercise Sheet 1 1. Prove that, for every δ > 0, there exists n N such that n - 1 < δ . 2. Let a 1 = 1 and a n +1 = a n a n + 2 for all n 2. Prove that the sequence is decreasing and find its limit. 3. Making references to any theorems about convergence which you use, evaluate lim n →∞ 2 n 3 + n 2 + n - n 5 n 3 + 2 n - 1 + 2 n 5 and lim n →∞ 2 n + 3 3 n + 2 . 4. Let k R and b > 1. Prove that lim n →∞ n k
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