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PRINT Your Name:
Quiz for June 8, 2004
Prove that if the sequence cfw_an converges to a , then the sequence cfw_an 
converges to a . Is the converse true? Prove or give a counter example.
ANSWER: Given > 0 , there exists n0 such that whenever n0 n ,
PRINT Your Name:
Quiz for June 5, 2006
2
+1
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2
given > 0 , you are required to nd a formula for n0 with the property that:
()
whenever n > n0 , then:
n2 + 1 1
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2n2
2
PRINT Your Name:
Quiz for June 1, 2006
Let X , Y , and Z be sets. Suppose that f : X Y and g : Y Z are
onetoone functions. Prove that the function g f : X Z is onetoone.
ANSWER: Let x1 and x2 be elements of X with (g f )(x1 ) = (g f )(x2 ) . We
must p
PRINT Your Name:
Quiz for June 6, 2005
Let cfw_pn be a bounded sequence of real numbers and let p R be such that
every convergent subsequence of cfw_pn converges to p . Prove that the sequence
cfw_pn converges to p .
ANSWER: Suppose that
(6)
the sequen
PRINT Your Name:
Quiz for June 15, 2004
Let cfw_an and cfw_bn be Cauchy sequences. Prove that the sequence cfw_an bn is also
a Cauchy sequence.
ANSWER: Let > 0 be arbitrary, but xed.
The sequence cfw_an is a Cauchy sequence; so, there exists n1 such
Math 554, Final Exam, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for the staple. Be sure to number your pages. Put your
Math 554, Final Exam, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for the staple. Be sure to number your pages. Put your
Math 554, Exam 1, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for the staple. Be sure to number your pages. Put your sol
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Math 544 Exam 4 Summer 2000
Use the paper provided. Put your name on the front of the rst page and the back
of the last page. Problem 7 is worth 8 points each. The other problems are worth
7 points each.
1. Dene open set.
2. Dene compact set.
3. Let f : E
Math 554, Exam 2, Summer 2006 Solutions
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for the staple. Be sure to number your pages. Pu
Math 554, Exam 2, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for the staple. Be sure to number your pages. Put your sol
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Notes on Exam 2, Math 554, Summer 2006
1. Exam 2 is Tuesday June 20 and it covers sections 2.1, 2.2, 2.3, 2.4, 2.6, 3.1, and
3.2.
2. Be able to DEFINE sequence, the sequence converges, the limit of a
sequence, monotone increasing, monotone decreasing, mon
Notes on Exam 3, Math 554, Summer 2005
1. Exam 3 is Thursday June 16 and it covers sections 2.1, 2.2, 2.3, 2.4, 2.6, and
3.1.
2. Be able to dene sequence, the sequence converges, the limit of a
sequence, monotone increasing, monotone decreasing, monotone,
Math 554, Exam 3, Summer 2005
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to problem 1 rst, and then your solution to number 2, etc; altho
Math 554, Exam 2, Summer 2005 Solutions
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to problem 1 rst, and then your solution to number 2,
Math 554, Exam 2, Summer 2005
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to problem 1 rst, and then your solution to number 2, etc; altho
Math 554, Exam 3, Summer 2005, solution
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to problem 1 rst, and then your solution to number 2,
Math 554, Exam 4, Summer 2005
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Be sure to number your pages. Put your solution
to problem 1 rst, and then your solution to number 2, etc; altho