ANALYSIS I: LAST PROBLEM SET (SNIFF!)
DUE FRIDAY, 6 MAY
Exercise 73. Which subsets of a metric space X can be written as the closure of their interior?
Exercise 74. Suppose {W j } j ∈N a cover of R (i.e., a collection of closed subsets whose union is R). Show that at least
one of the W j has nonempty interior.
Exercise 75. Suppose K ⊂ R a compact subset, and suppose W ⊂ R a closed subset, disjoint from K. Show that
there is a real quantity κ > 0 such that for any x ∈ K and any y ∈ W , one has |x − y| > κ. Show that this is false if
we only assume K closed, or if we assume that both K and W are open.
Exercise 76. Suppose (an )n≥0 and (bn )n≥0 two sequences of real numbers. Which of the following is greater? In each
case, give examples when equality obtains, as well as examples when it does not.
lim sup(an + bn ) and
lim sup an + lim sup bn ;
lim sup(an bn ) and
(lim sup an )(lim sup bn );
n≥0
n≥0
n≥0
lim sup an−1
n≥0
n≥0
n≥0
n≥0
(lim sup an ) ;
−1
and
n≥0
lim inf(an + bn ) and
lim inf an + lim inf bn ;
lim inf(an bn ) and
(lim inf an )(lim inf bn );
n≥0
n≥0
n≥0
lim inf an−1
n≥0
n≥0
n≥0
n≥0
(lim inf an )−1 .
and
n≥0
Exercise 77. What are the limit points of the sequence (cos n)n≥0 ?
Exercise 78. Let S denote the set
{1/n | n ≥ 1 an integer} ∪ {0}.
Find a sequence (xn )n≥0 whose set of limit points is precisely S.
Exercise? 79. Consider the sequence (xn )n≥1 with
xn := − log n +
n 1
X
k=1
k
.
Prove that this sequence converges by writing γ := lim supn≥1 xn and finding a number c > 0 such that |xn −γ | ≤ c/n
for any n ≥ 1.
P
P
Exercise 80. Suppose (an )n≥0 a sequence of positive real numbers. Prove that ∞
a converges only if ∞
a2
n=0 n
n=0 n
converges. Give an example to show that the positivity condition is necessary.
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Exercise 81. Suppose (an )n≥0 a sequence of positive real numbers. Prove that if ∞
a converges, then
n=0 n
diverges, and
converges.
∞
X
1
n=0
1 + an
∞
X
an
n=0
1 + an
1
2
DUE FRIDAY, 6 MAY
Exercise 82. Does the series
∞ (2 j + 3)1/2 − (2 j )1/2
X
j 1/2
j =1
converge?
Exercise 83. Suppose (an )n≥1 a sequence of real numbers. For any N ≥ 1, let αN be the average (arithmetic mean)
of a1 , a2 , . . . , aN . Show that if (an )n≥1 converges, then (αN )N ≥1 converges to the same limit. Give a counterexample
to the converse.
Exercise? 84. Suppose (an )n≥0 and (bn )n≥0 two sequences of real numbers. Suppose b0 ≥ b1 ≥ b2 ≥ · · · , and
P
limn→∞ bn = 0. Show that if the partial sums AN = Nj=0 ak form a bounded sequence, then the series
∞
X
ak bk
k=0
converges.
Exercise 85. Suppose f a real-valued function on R. If A, B ⊂ R are disjoint, does it follow that f (A) and f (B) are
disjoint? How about f −1 (A) and f −1 (B)?
Y and a connected subset E ⊂ Y such that f −1 E
Exercise 86. Give an example of a continuous function f : X
Y and a compact subset K ⊂ Y such that
is not connected. Give an example of a continuous function g : X
g −1 K is not compact.
Exercise 87. Suppose X a metric space, and suppose S ⊂ X . Define a real valued function f on X by the formula
f (x) := inf{d (s, x) | s ∈ S}.
Is f uniformly continuous?
Exercise 88. Suppose E, F ⊂ R two disjoint closed subsets. Show that there is a real-valued continuous function f
on R such that f −1 (0) = E and f −1 (1) = F .
Exercise 89. Suppose K, L ⊂ Rn . Write
K + L := {x + y | x ∈ K and y ∈ L}.
If K and L are compact, does it follow that K + L is compact as well?
Exercise 90. Suppose f a real-valued function on an open interval I ⊂ R. Suppose that f is convex, in the sense that
for any x, y ∈ I and any t ∈ [0, 1], then
f ((1 − t )x + t y) ≤ (1 − t ) f (x) + t f (y).
Prove that f is continuous.
Exercise 91. Suppose f a real-valued function that is continuous on [0, ∞) and differentiable on (0, ∞). Show that
if f (0) = 0 and | f 0 (x)| ≤ | f (x)| for any x > 0, then f = 0.
Exercise 92. Suppose f a differentiable real-valued function on an open interval I ⊂ R. Show that f 0 is continuous
if and only if the inverse image of any point is closed.
Exercise 93. Suppose f a differentiable real-valued function on R, and suppose f 0 never vanishes. Show that f is a
homeomorphism, show that f −1 is differentiable, and find an expression for ( f −1 )0 .
Exercise 94. Show that the improper integral
Z
∞
0
sin x
x
exists; that is, show that the limits
Z
lim lim
b →∞ "→0 "
exist and are equal.
b
sin x
x
Z
and
lim lim
"→0 b →∞ "
b
sin x
x
ANALYSIS I: LAST PROBLEM SET (SNIFF!)
3
Exercise 95. Must the composition of two Riemann integrable functions [a, b ]
[a, b ] be Riemann integrable?
Exercise 96. Suppose f a real-valued function on [a, b ]. For any x ∈ [a, b ], define the total variation


k
X

V f[a,b ] := sup
| f ( p j ) − f ( p j −1 )| | a = p0 ≤ p1 ≤ · · · ≤ pk = b .


j =0
Show that if f is continuously differentiable on [a, b ], then
Zb
V f[a,b ] =
| f 0 (x)| dx.
a
Exercise 97. Suppose f a continuous real-valued function on the closed interval [0, 1] with f ≥ 0. Show that
‚Z 1
Œ1/n
n
lim
f (x) dx
= sup f (x).
n→∞
x∈[0,1]
0
Exercise 98. Consider the function α(x) = x − bxc. Compute
Z5
x dα.
0
?
Exercise 99. Suppose f ∈ C ([0, 2π]) such that f (0) = f (2π) and f 0 (0) = f 0 (2π). For any natural number n,
define
Z
1 2π
f (x) sin(n x) dx.
fb(n) :=
2π 0
Show that the series
∞
X
| fb(n)|2
1
n=1
converges.
Exercise 100. A real-valued function f on [a, b ] is said to be polygonal if is continuous, and if there exists a partition
a = p0 ≤ p1 ≤ · · · ≤ pn = b such that f is affine on each [ pi , pi +1 ]. Show that the set of polygonal functions is dense
in the space of continuous functions on [a, b ].
Exercise 101. Show that sin and cos are not polynomials.
Exercise 102. Prove that if f is a continuous real-valued function on [a, b ] such that for any polynomial p, one has
Zb
f (x) p(x) dx = 0,
a
then f = 0.
Exercise 103. Prove that the sequence of real-valued functions ( f j ) j ≥0 on R defined by f j (x) = sin( j x) has no
convergent subsequence.
Exercise 104. Suppose ( f j ) j ≥0 a pointwise convergent sequence of real-valued functions on a compact space K such
that for any j ≥ 0 and any x, y ∈ K, one has
d ( f j (x), f j (y)) ≤ d (x, y).
Show that ( f j ) j ≥0 converges uniformly.
Exercise? 105. Prove that Euler’s constant e is transcendental in the following manner. Suppose a0 + a1 e + · · · +
a m x e m = 0 for some integer coefficients ai with a0 6= 0. For any prime number p, write
g (x) :=
x p−1 (x − 1) p (x − 2) p · · · (x − m) p
( p − 1)!
,
4
DUE FRIDAY, 6 MAY
and set
G(x) :=
m p+
Xp−1
g (k) (x).
k=0
Check that
| g (x)| <
m m p+ p−1
( p − 1)!
,
d
dx
(e
−x
G(x)) = −e
−x
Z
g (x),
Deduce that
m
X
j =0
aj e
j
Z
j
e
−x
g (x) dx = −
0
aj
and
m m p+
X
Xp−1
j =0
i =0
0
j
e −x g (x) dx = a j G(0) − a j e − j G( j ).
a j g (i ) ( j ).
(i )
Now show that g ( j ) is always an integer, and show that it is divisible by p unless j 6= 0 and i 6= p − 1. If p > m,
show that g ( p−1) (0) is not divisible by p. Finally, show that if p is sufficiently large, then
−
m m p+
X
Xp−1
j =0
a j g (i) ( j )
i=0
is nonzero, yet


X
Z j
m+2 p−1
m
X
m
)
 m (m
−x
j
≤
e
|a
|
e
g
(x)
dx
a
e
< 1,
j 
j

( p − 1)!
j =0
0
j =0
a contradiction.