Spring 2010: MATH 3210-001
Foundations of Analysis I
Wed. 10th March
Midterm Examination
Student ID:
Name:
Instructions.
1. Attempt 4 out of the 5 questions.
2. Circle the 4 question numbers which you want to be graded.
3. Give the full statement of a theorem when asked to state it.
4. If you need a theorem to justify your answer, then you must give the full statement.
5. Scratch paper is on the last page. Ask the instructor if you need some more.
Question Points Your Score
Q1
10
Q2
10
Q3
10
Q4
10
Q5
10
TOTAL
40
Dr. R. Lodh
50 minutes
Q1]. . . [10 points] Let {an } be a sequence of real numbers
(i) Define what it means to say that {an } is convergent.
(ii) Prove that if {an } is convergent, then it is bounded.
OVER
Q2]. . . [10 points] Let {an } and {bn } be convergent sequences with an → a and bn → b.
(i) Prove that {an + bn } is a convergent sequence and an + bn → a + b.
(ii) Prove that if an > 0 for all n, then a ≥ 0.
Q3]. . . [10 points]
(i) State and prove the Monotone Convergence Theorem.
(ii) Let {an } be the sequence defined by a1 = 1 and an+1 = (1 −
converges.
1
)a
n+1 n
for n > 1. Show that {an }
Q4]. . . [10 points] Let D ⊂ R and f : D → R a function.
(i) Define what it means to say that f is continuous at a ∈ D. Then define what it means to say that f
is continuous on D.
(ii) Is the function f : [−1, 1] → R defined by
(
1/x −1 ≤ x < 0
f (x) =
0
0≤x≤1
continuous? Justify your answer.
Q5]. . . [10 points]
(i) State the Intermediate Value Theorem.
(ii) Suppose that f (x) = x10 + x3 − a for some real number a > 0. Prove that f (x) has a root, i.e. that
there is real number b such that f (b) = 0.
Scratch Paper