University of Ottawa
Department of Mathematics and Statistics
MAT 3520 : Real Analysis
Professor : Alistair Savage
Midterm Test
20 October 2016
Surname
First Name
Student #
Instructions :
(a) You have 80 minutes to complete this exam.
(b) The number of points available for each question is indicated in square brackets.
(c) Unless otherwise indicated, you must justify your answers to receive full marks.
(d) All work to be considered for grading should be written in the space provided. The
reverse side of pages is for scrap work. If you find that you need extra space in order
to answer a particular question, you should continue on the reverse side of the page
and indicate this clearly. Otherwise, the work written on the reverse side of pages will
not be considered for marks.
(e) Write your student number at the top of each page in the space provided.
(f) No notes, books, scrap paper, calculators or other electronic devices are allowed.
(g) You may use the last page of the exam as scrap paper.
Good luck !
Please do not write in the table below.
Question
Maximum
Grade
1
2
2
6
3
6
4
4
5
4
6
4
Total
26
Student #
MAT 3520 Midterm Test
Question 1. [2 points] Let (X, d) be a metric space and let F ⊆ X be a finite subset.
Prove that F is closed in X.
Question 2.
(a) [1 point] Write down the definition of the metric space `p (p ≥ 1). (You do not need
to prove that it is a metric space.)
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MAT 3520 Midterm Test
(b) [5 points] Prove that `1 is complete.
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MAT 3520 Midterm Test
Question 3.
(a) [2 points] Let {fn }∞
n=1 be a sequence of functions from R to R (i.e. fn : R → R for
all n ∈ N+ ). Show that if the sequence converges uniformly to a function f : R → R, then it
converges pointwise to f .
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MAT 3520 Midterm Test
(b) [4 points] Give an example of a sequence of functions {fn }∞
n=1 , where fn : R → R for
all n ∈ N+ , such that {fn } converges pointwise but it does not converge uniformly. Justify
your answer. That is, prove that your sequence converges pointwise but that it does not
converge uniformly.
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MAT 3520 Midterm Test
Question 4. [4 points] Let X be a nonempty set and let F be the set of all finite subsets
of X. Show that the function
d : F × F → R≥0 ,
d(A, B) = |A∆B|,
defines a metric on F , where |Y | denotes the number of elements of a finite set, and A∆B =
(A \ B) ∪ (B \ A) is the symmetric difference of A and B.
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MAT 3520 Midterm Test
Question 5.
(a) [1 point] Write down the definition of a sequentially compact metric space. (You do
not need to give the definition of a metric space.)
(b) [3 points] Suppose that (X, dX ) and (Y, dY ) are sequentially compact metric spaces.
Then the function
d : (X × Y ) × (X × Y ) → R≥0 ,
d((x1 , y1 ), (x2 , y2 )) = dX (x1 , x2 ) + dY (y1 , y2 ),
defines a metric on X × Y . (You do not need to prove this.) Prove that the space (X × Y, d)
is sequentially compact.
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MAT 3520 Midterm Test
Question 6.
(a) [1 point] Write down the definition of a connected topological space. (You do not
need to give the definition of a topological space.)
(b) [3 points] Let X be an infinite set and let
T = {A ⊆ X | A = ∅ or X \ A is finite}.
Then T is a topology on X. (You do not need to prove this.) Show that X is connected.
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MAT 3520 Midterm Test
This page is intentionally left blank. You may use it as scrap paper.
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