1. Let (Jn) be a nested sequence of open intervals: J2 J2 2 J3 2.... Is...

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∞ 1. Let (Jn) be a nested sequence of open intervals: J₁2 J2 2 J3 2.... Is J, always nonempty? If your answer is yes, give a proof; if your answer is no, give a counterexample. (Hint: consider an exercise from the last homework.) n=1 2. Let (Jn) be a strictly shrinking nested sequence of open intervals. i.e. Jn = (an, bn), an <bn and an <an+1, bn > bn+1 for all n € N. Prove that Jn #o. (Hint: connect to the nested interval property for closed intervals.) 3. Prove the following limits by definition. (a) lim n non +3 1 n=s (1.1)" 1 (b) lim (c) lim x ². (d) lim -n n = = 0. 1. = 0. 818 n²+1 1. n=1 4. Let C> 0 be a constant. Prove that lim = L if and only if the following statement is true: 87x For every > 0, there exists NEN such that whenever n > N, it is true that xn-L<Ce. 5. Prove that lim n = L if and only if the following statement is true: 1148 For every > 0, there exists NEN such that whenever n > N, it is true that xn-L≤e. ∞ 1. Let (Jn) be a nested sequence of open intervals: J₁2 J2 2 J3 2.... Is J, always nonempty? If your answer is yes, give a proof; if your answer is no, give a counterexample. (Hint: consider an exercise from the last homework.) n=1 2. Let (Jn) be a strictly shrinking nested sequence of open intervals. i.e. Jn = (an, bn), an <bn and an <an+1, bn > bn+1 for all n € N. Prove that Jn #o. (Hint: connect to the nested interval property for closed intervals.) 3. Prove the following limits by definition. (a) lim n non +3 1 n=s (1.1)" 1 (b) lim (c) lim x ². (d) lim -n n = = 0. 1. = 0. 818 n²+1 1. n=1 4. Let C> 0 be a constant. Prove that lim = L if and only if the following statement is true: 87x For every > 0, there exists NEN such that whenever n > N, it is true that xn-L<Ce. 5. Prove that lim n = L if and only if the following statement is true: 1148 For every > 0, there exists NEN such that whenever n > N, it is true that xn-L≤e.

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No a counterexample is Jn 1n 2n for n 1 Let In an bn be ... View the full answer