Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Exam #1
Due: Tuesday, June 9
This is a take-home examination. The exam includes 8 questions. The total mark is 100
points. Please show all the work, not only the answers.
1. [12 points] Fix any d ∈ N and let k · k denote the usual Euclidean norm in Rd . That is,
v
u d
uX
kxk = t
x2i for a vector
x = (x1 , . . . , xd ) ∈ Rd .
i=1
Suppose that u, v ∈ Rd . Find w ∈ Rd and r > 0 such that
kx − uk = 2kx − vk
if and only if kx − wk = r.
Hint: Prove that w =
1
3
4v − u and r = 23 v − u.
2. [14 points] Solve Exercise 35 in Chapter 1 of the textbook.
3. [12 points] Let E be the set of all real numbers x ∈ [0, 1] whose decimal expansion
contains only the digits 4 and 7. Is E countable?
4. [14 points] Fix any real number α > 1. Take x1 >
xn+1
√
α − x2n
α + xn
= xn +
,
=
1 + xn
1 + xn
α and define recursively
n ∈ N.
(a) Prove that x1 > x3 > x5 > . . . .
(b) Prove that x2 < x4 < x6 < . . . .
√
(c) Prove that limn→∞ xn = α.
√
(d) Let εn = |xn − α|. Show that εn < cβ n for some constants c > 0 and β ∈ (0, 1).
5. [12 points] Fix any d ∈ N and let k · k denote the usual Euclidean norm in Rd . That is,
v
u d
uX
kxk = t
x2i for a vector
x = (x1 , . . . , xd ) ∈ Rd .
i=1
1
We say that a sequence (xn )n∈N in Rd converges to x ∈ Rd and write limn→∞ xn = x if for
any ε > 0 there exists Nε ∈ N such that
n ∈ N and m, n > Nε
⇒
kxn − xk < ε.
Call two converging sequences X = (xn )n∈N and Y = (yn )n∈N in Rn equivalent and write
X ∼ Y if
lim kxn − yn k = 0.
n→∞
Let E be the set of all converging sequences in Rn . Show that ∼ is an equivalence relation
in E.
6. [12 points]
(a) Let (sn )n∈N be a sequence of reals such that
sn+1 =
sn + sn−1
.
2
Show that sn is a Cauchy sequence and hence converges.
(b) Let (sn )n∈N be a sequence of reals defined recursively by
s1 = 0, ,
s2n =
s2n−1
,
2
s2n+1 =
1
+ s2n .
2
Find lim supn→∞ sn and lim inf n→∞ sn .
Hint: Consider the sequences un = s2n and vn = s2n−1 separately.
7. [12 points] Let (sn )n∈N be a sequence of reals and define
n
1X
tn =
si .
n i=1
(a) Prove that if limn→∞ sn = s then limn→∞ tn = s.
(b) Give an example to show that tn can converge even though sn doesn’t.
8. [12 points]
(a) Use induction to show that if (x + 1/x) is integer then (xn + 1/xn ) is also integer for
any n ∈ N.
(b) Show that
sup(A ∪ B) = max{sup A, sup B}
for any sets A, B ⊂ R.
2