Sample Midterm Problems Math 3210
Spring 2010
Instructor: Rémi Lodh
Mar. 1st 2010
NB: When you are asked to state a theorem, then you must give the full statement
of the theorem. Similarly, if you need a theorem to justify your answer, then you must
give the full statement of the theorem.
1. Let {an } be a sequence of real numbers.
(i) Define what it means to say {an } is convergent.
(ii) Define what it means to say that {an } is a Cauchy sequence.
(iii) Prove that if {an } is convergent, then it is a Cauchy sequence.
2. Prove that if a sequence {an } converges, then it is bounded.
3. Prove that if lim an = a, then lim |an | = |a|.
4.
(i) State and prove the Squeeze Theorem.
(ii) Show that lim
sin(n)
= 0.
n
5. Prove that if {an } and {bn } are convergent sequences with an → a and bn → b,
then an + bn → a + b.
6.
(i) State and prove the Monotone Convergence Theorem.
(ii) Show that the sequence {an } defined a1 = 1 and an+1 = (1 −
gent.
7.
(i) State the Monotone Convergence Theorem.
(ii) State and prove the Nested Interval Property.
8.
(i) State the Bolzano-Weierstrass Theorem.
(ii) Use it to prove that a Cauchy sequence is convergent.
9. Let D ⊂ R and f : D → R a function.
1
1
n+1 )an
is conver-
(i) Define what it means to say that f is continuous at a ∈ D.
(ii) Use your definition to prove that the function
f :R
→ R
x
7→ x2
is continuous at x = 0.
10. Let f, g : D → R be functions on D.
(i) State the criterion for continuity using convergent sequences.
(ii) Prove that if f and g are both continuous at a ∈ D, then so is f + g. State
clearly any theorem you may need for your proof.
11. Is the function f : [−1, 1] → R defined by f (x) = 1/x for −1 ≤ x < 0 and
f (x) = 0 for 0 ≤ x ≤ 1 continuous? Justify your answer.
12.
(i) State the Intermediate Value Theorem.
(ii) Suppose f (x) = x6 + x2 − a, where a is a positive real number. Prove that f
has a root, i.e. that there is a real number b such that f (b) = 0.
13. Let f : D → R be a function.
(i) Define what it means to say that f is uniformly continuous on D.
(ii) Consider the function
f : (−1, 1] →
x 7→
R
1
.
x+1
Is f uniformly continuous on (−1, 1]? Justify your answer.
http://www.math.utah.edu/~remi/teaching/3210Spr2010/3210Spr2010.html
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