# 1 MS Exam’n — Analysis — Jan. 15, 2002 Name: ```MS Exam’n — Analysis — Jan. 15, 2002 Name:
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Problem 1. Suppose that E is an uncountable subset of R. Let the set M
be defined by
M = { x ∈ R | (−∞, x] ∩ E is finite or countable} .
Prove that if M 6= ∅, then M is bounded above Also show that if b = sup M ,
then b ∈ M , and for some x, x &gt; b, both sets (−∞, x] ∩ E and [x, +∞) ∩ E
are uncountable, still assuming that M 6= ∅.
MS Exam’n — Analysis — Jan. 15, 2002 Name:
Blank Page
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MS Exam’n — Analysis — Jan. 15, 2002 Name:
Problem 2. Suppose that {an } is a decreasing sequence of non-negative
∞
X
an converges
real numbers: an ≥ an+1 ≥ 0 for n = 1, 2, . . . . Prove that if
n=1
then lim nan = 0.
n→∞
MS Exam’n — Analysis — Jan. 15, 2002 Name:
Blank Page
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MS Exam’n — Analysis — Jan. 15, 2002 Name:
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Problem 3. Suppose that f : [0, +∞) →Z R is a continuous function such
1 x
f (t) dt = 0.
that lim f (x) = 0. Prove that lim
x→+∞
x→+∞ x 0
MS Exam’n — Analysis — Jan. 15, 2002 Name:
Blank Page
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MS Exam’n — Analysis — Jan. 15, 2002 Name:
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Problem 4. Suppose that {fn } is a sequence of continuous functions on
the interval [0, 1] which converges pointwise to 0 on [0, 1]; that is for each
x ∈ [0, 1]
lim fn (x) = 0.
n→∞
Show that if, for each x ∈ [0, 1], the sequence {f n (x)} is decreasing, then
{fn } converges uniformly to 0 on the interval [0, 1].
MS Exam’n — Analysis — Jan. 15, 2002 Name:
Blank Page
mscompanal02.dvi
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