EXAM
Practice Problems for Exam 1
Math 4350, Fall 2009
October 27, 2009
• These are practice problems to give you an idea of
what will be on the exam. Of course, there are more
problems here than I could put on an inclass exam.
• This exam has 14 problems.
Good luck!
Problem 1. State the Well-ordering Property of N.
Problem 2. Prove by induction that
1
1
1
1
+ ··· + n = 1 − n.
+
2 22
2
2
Problem 3. Define the supremum of a nonempty set of real numbers. State
the The Completeness Property of R.
Problem 4. Let A be a nonempty subset of R which is bounded above. Show
that a number s is the supremum of A if and only if the following
two statements hold:
A. s is an upper bound for A.
B. For every ε > 0, there is an a ∈ A so that s − ε < a.
Problem 5. Show that
inf{1/n2 | n ∈ N} = 0.
Problem 6. Let A and B be nonempty subsets of R that are bounded above.
Define
A + B = {a + b | a ∈ A and b ∈ B}.
Show that sup(A + B) exists and
sup(A + B) = sup(A) + sup(B).
Problem 7. Let (xn ) be a sequence in R. State the definition of
lim(xn ) = L,
where L ∈ R.
Problem 8. Show that a convergent sequence of real numbers is bounded.
1
Problem 9. Let (xn ) be a decreasing sequence of real numbers which is bounded
below. Show that
lim(xn ) = inf{xn | n ∈ N}.
Problem 10. Let (xn ) be a convergent sequence of real numbers. Suppose that
xn ≤ b for all n ∈ N. Show that
lim(xn ) ≤ b.
Give an example where xn < b for all n, but lim(xn ) = b.
Problem 11. Define a sequence as follows. Let x1 = 1 and define the sequence
recursively by
xn+1 =
1
(xn + 1).
3
Show that (xn ) is convergent and find the limit. (Hint: show xn ≥ 1/2).
Problem 12. Let
∞
X
an
n=1
be a series of real numbers. Prove the following two statements.
A. If the series converges, then lim(an ) = 0.
B. If lim(an ) does not exist, or if lim(an ) exists but is not zero, then the series
diverges.
Problem 13. Determine if the series
∞
X
n=1
√
n+1
+ n2 + 8
n3
is convergent or divergent. Justify your answer, of course.
2
Problem 14. You may assume the existance of square roots.
√
A. Show that 2 is irrational.
√
B. Let
√ a be a natural number. Show that if a is not a natural number, then
a is irrational. This implies the first part, of course.
3