MA 452/502: Introduction to Real Analysis
Final Exam
Name: Score: /100
December 03, 2007
There are 7 problems in this exam for a maximum of 100 points. Please show
your work and present your solutions in a well organized way. No work, no credit.
1. (25 points) Disprove the following statements by constructing counterexamples. Justify
your answer please.
(1). If a nonempty bounded set S⊆Rcontains its maximum and minimum, then Sis
compact.
(2). Every unbounded sequence has no convergent subsequences.
(3). If limx→cf2(x) = L > 0, then limx→cf(x) exists and is finite.
(4). Let f:I→Rand g:I→R. If fand gare uniformly continuous on I, then the
product function f g is uniformly continuous on I.
(5). Let f(x) and g(x) be integrable on [a, b], and let f(x)≤h(x)≤g(x) for all x∈[a, b],
then h(x) is integrable on [a, b].
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