Question

8. Show that Theorem 3.1, the Nested intervals theorem, may be proved as a direct consequence of the Cauchy criterion for convergence (Theorem 3.14). (Hint: Suppose I. = {x: 0, <x<bn} is a nested sequence. Then show that {an} and {b} are Cauchy sequences. Hence they each tend to a limit. Since b.-4, 0, the limits must be the same. Finally, the Sandwiching theorem shows that the limit is in every 1.]

Answer 1

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