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6.1 1. (a) In Example 6.1, give another proof that cn + L by using the result of Exercise 5.4/1. (b) Given a set of nested intervals (an, bn), n = 0,1,2, ..., for which bn an + 0, prove the sequence ao, bo, 21,b, 22, b2, ..., An, bn, ... converges. (This generalizes part (a) and can be done the same way.) 2. Let a be a real number between 0 and 1 written in binary: e.g., a = .1011001... a = 1/2+1/23 + 1/24 + 1/2? +.... Make a set of nested intervals by starting with lo = {0, 1), then defining recursively In to be the (closed) left half of In-1 if the n-th place of a is 0, and the (closed) right half if the n-th place is 1. Prove the resulting sequence of nested intervals converges to a, i.e., a is the unique number inside all the intervals. means 6.1 1. (a) In Example 6.1, give another proof that cn + L by using the result of Exercise 5.4/1. (b) Given a set of nested intervals (an, bn), n = 0,1,2, ..., for which bn an + 0, prove the sequence ao, bo, 21,b, 22, b2, ..., An, bn, ... converges. (This generalizes part (a) and can be done the same way.) 2. Let a be a real number between 0 and 1 written in binary: e.g., a = .1011001... a = 1/2+1/23 + 1/24 + 1/2? +.... Make a set of nested intervals by starting with lo = {0, 1), then defining recursively In to be the (closed) left half of In-1 if the n-th place of a is 0, and the (closed) right half if the n-th place is 1. Prove the resulting sequence of nested intervals converges to a, i.e., a is the unique number inside all the intervals. means

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