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are 1 open intervals having total length 7 10 55 a

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Unformatted text preview: S be a set of real numbers having measure zero. ∞ Prove that, for any given ǫ &gt; 0, the set S ⊆ In , where the In , n = 1, 2, 3, . . . are 1 open intervals having total length &lt; ǫ . 7. (10: 5,5) a) “If a sequence {xn }, n ≥ 0 is bounded above for n ≫ 1, then it is bounded above for all n ≥ 0.” The analogous statement for a function f (x) deﬁned on [0, ∞] would be: “If f (x) is bounded above for x ≫ 1, it is bounded above for x ≥ 0 .” Show that this statement is false; strengthen the hypothesis on f (x) in a reasonable way, and prove the amended statement. � b) “If an inﬁnite series an converges, then lim an = 0.” An analogous statement for n→∞ functions would be: �∞ “If f (x) is continuous on [0, ∞), and 0 f (x) dx converges, then lim f (x) = 0. n→∞ Show this statement is false; strengthen the hypothesis on f (x) so the amended statement is true. (You need not prove it.) 8. (10: 6, 4) Let f (x) be uniformly continuous on the interval I in R; the interval I is not assumed to be compact. a) Prove that if {xn } is a Cauchy sequence in I, then {f (xn )} is a Cauchy sequence. b) Use part (a) to prove that f (x)...
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This note was uploaded on 01/22/2014 for the course MATH 18.100A taught by Professor Arthurmattuck during the Fall '12 term at MIT.

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