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Unformatted text preview: S be a set of real numbers having measure zero.
∞ Prove that, for any given ǫ > 0, the set S ⊆
In , where the In , n = 1, 2, 3, . . . are
1 open intervals having total length < 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.
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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.
 Fall '12
 ArthurMattuck

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