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  3. 2 prove that the interval a b is not a...

Question: 2 prove that the interval a b is not a...

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*2. Prove that the interval [a, b is not a zero set. (a) Explain why the following observation is not a solution to the problem Every open interval that contains [a,b] has length 〉 b-a. (b) Instead, suppose there is a bad covering of [a, b by open intervals {I, whose total length is 〈 b-a, and justify the following steps. (i) It is enough to deal with finite bad coverings (ii) Let B {1.. . . , In} be a bad covering such that n is minimal among all bad coverings (iii) Show that no bad covering has n-1 so we have n 〉 2 (iv) Show that it is no loss of generality to assume a E 11 and 11 「メ1 (v) Show that 1-11 U 12 is an open interval and |1|く|11| + 21 (vi) Show that B- {I,I3, , Im) is a bad covering of [a, b with fewer intervals, a contradiction to minimality of n

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