Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Exam #1
Sample
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
x2i for a vector
x = (x1 , . . . , xd ) ∈ Rd .
kxk = t
i=1
Suppose that u, v ∈ Rd and ku − vk > 1. Find all x ∈ Rd such that
kx − uk2 = kx − vk2 + 1.
2. [12 points] Consider the usual Euclidean spaces R2 equipped with the polar
p coordinates
(r, θ). Thus, as usual, if x = (x1 , x2 ) in the cartesian coordinates, then r = x21 + x22 and
tan θ = x2 /x1 , θ ∈ [0, 2π) unless x1 = 0. If x1 = 0 then θ is either π/2 or 0 or 3π/2, depending
on the sign of x2 .
Call two non-zero vectors x = (rx , θx ) and y = (ry , θy ) (written in the polar coordinates)
equivalent and write x ∼ y if θx = θy .
(a) Show that ∼ is an equivalence relation in R2 \{0}.
(b) Let E denote the set of equivalence classes induced by ∼ in R2 \{0}. Verify that the
elements of E are infinite arrays (open half-lines) starting at zero (the origin itself is
not included).
(c) We can identify the elements in E (arrays) with the corresponding values of θ. Thus,
we identify θ ∈ [0, 2π) with the array
x = (x1 , x2 ) ∈ R2 : x1 = t cos θ and x2 = t sin θ for some t ∈ (0, ∞) .
Correspondingly, we introduce a distance in [0, 2π) by setting
p
d(θ, ϕ) = (cos θ − cos ϕ)2 + (sin θ − sin ϕ)2 .
Give a geometric interpretation of d(θ, ϕ) in R2 .
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(d) Show that the distance d(·, ·) satisfies the triangle inequality, that is for any θ, ϕ, ψ ∈ [0, 2π),
d(θ, ϕ) ≤ d(θ, ψ) + d(ψ, ϕ).
When the equality holds?
3. [12 points]
(a) Let E be the set of all real numbers x ∈ [0, 1] whose decimal expansion contains only
the digits 3, 4, and 5. Is E dense in [0, 1]? Recall that a set is called dense in [0, 1] if
its intersection with any (open) subinterval of [0, 1] is non-empty.
(b) Let N denote the set of all positive integers, i.e. N = {1, 2, 3, . . .}. Let A and B be two
(not necessarily bounded) subsets of N, and define
C = {z ∈ N : there exist x ∈ A and y ∈ B such that z ≤ x + y}.
Prove that sup C = sup A + sup B.
√
4. [14 points] Fix any real number α > 0. Choose x1 > α and define recursively
1
α
xn+1 =
xn +
,
n ∈ N.
2
xn
√
(a) Prove that xn is a monotone decreasing sequence and that lim→∞ xn = α.
√
(b) Put εn = xn − α and show that
ε2n
ε2n
< √ .
εn+1 =
2xn
2 α
√
n
Conclude that εn+1 < β(ε1 /β)2 , where β = 2 α, and thus the above recursion provides a fast algorithm for computing square roots.
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
A sequence (xn )n∈N of points in Rd is called Cauchy if for any ε > 0 there exists Nε ∈ N
such that
m, n ∈ N and m, n > Nε
⇒
kxn − xm k < ε.
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.
Prove that a sequence in Rn converges if and only if it is a Cauchy sequence.
6. [14 points]
2
(a) Let f (x) be a real-valued function such that
|f (x) − f (y)| <
5
· |x − y|
6
for all x, y ∈ R. Prove that the sequence defined by xn+1 = f (xn ) converges for any
x0 ∈ R.
Hint: Using the above inequality, prove by induction that
5 n
· |x1 − x0 |,
n ∈ N.
|xn+1 − xn | <
6
Then use the triangle inequality to evaluate |xn+m − xn | for arbitrary n, m ∈ N. Conclude that (xn )n∈N is a Cauchy sequence in R.
(b) Let (xn )n∈N be a sequence of reals and a ∈ R be a number. Show that xn converges to
a if and only if any it subsequence does.
7. [12 points]
(a) Let (sn )n∈N be a sequence of reals and s ∈ R be a number. Prove that if
sn − s
=0
n→∞ sn + s
lim
then limn→∞ sn = s.
(b) Introduce an order in the usual Euclidean space R2 as follows. If x = (x1 , x2 ) and
y = (y1 , y2 ) ∈ R2 we say that x < y if either x1 < y1 or x1 = y1 and x2 < y2 . Such
an order is often called lexicographic. Show that R2 ordered by the lexicographic order
doesn’t possess the least upper bound property, namely there is a bounded from above
set A ⊂ R2 for which sup A cannot be defined as an element of R2 .
8. [12 points]
(a) Use induction to show that 62n−1 + 1 divides 7 for any n ∈ N.
(b) Show that there is a sequence of reals (xn )n∈N such that for any x ∈ R there exists a
subsequence of xn converging to x. Note, you are not required to specify the values of
xn , just show that such a sequence exists.
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