Suppose that S is a nonempty set of real numbers that is bounded and that inf(S) = sup(S). Prove that the set S consists of exactly one number. Proof: S is a nonempty set of real numbers that is bounded above and below. Therefor by the Completeness A
Math534A - Fall 2016
Solutions Homework 10
Jrme Gilles
Exercise 1
We have
(cf )0 (x) = lim
h0
cf (x + h) cf (x)
c(f (x + h) f (x)
f (x + h) f (x)
= lim
= c lim
= cf 0 (x).
h0
h0
h
h
h
Exercise 2
We have
(f + g)(x + h) (f + g)(x)
h
f (x + h) + g(x + h) f (
Math534A - Fall 2016
Solutions Homework 8
Jrme Gilles
Exercise 1
First, we want to prove that limx2/3+
7
3x2
= +, i.e
M > 0, > 0, x R,
2
2
7
<x< +
> M.
3
3
3x 2
First note that we consider x > 32 . We want to get
M > 0,
7
7
7
2
7
> M 7 > M (3x 2) 3x 2 <
Math534A - Fall 2016
Solutions Homework 5
Jrme Gilles
Exercise 1
1.1
Note that 2 is an upper bound of S if
1
(1)n
2
n
(1)n+1 n.
Since (1)n+1 = 1 or 1, we have (1)n+1 1. Moreover, n 1 thus the expected condition holds so 2 is
an upper bound of S.
Next, we
Math534A - Fall 2016
Solutions Homework 6
Jrme Gilles
Exercise 1
We have, n N, n2 +
Therefore,
1
n
>
n
2
thus in order to get M > 0, n2 +
M > 0, N N, N > 2M, n N, n N,
1
n
n
2
> M , we can choose
1
n
+ > M lim
n
2
n
n
1
+
2
n
> M n > 2M .
= +.
Exercise 2
Math534A - Fall 2016
Solutions Homework 2
Jrme Gilles
Exercise 1
The triangle inequality is given by
a, b R, |a + b| |a| + |b|.
Let assume that a and b are of the same sign.
Case 1: a and b are both positive then |a| = a and |b| = b. We have a + b 0 thus
Math 534A Fall 2013
Problem Set # 3
Due on Tuesday, October 1
In problems 1-3, prove the assertion in accordance with the relevant precise definition:
1.
lim 23 = +
2
2
= +
+ 1
lim
3.
23
= +
2 4
lim
In problems 4 and 5, prove that is continuous at 0 in
Math 543A Fall 2013
Problem Set 3
Solutions
1. Since
1
2 = 2 2
3
and
2
we have
3
1
1 for each 1
2
1
23 = 3 2 2 3 3
if . Thus, given 0 we can pick
3 13
If then
Therefore,
23 3 3
lim 23 = +
as claimed.
2. We have
2
=
+1
1+
Since
1+
we have
1
1
1+1=2
1+
An Alternative (and better) Solution for Problem 9 in Problem Set 3
(as we did in class)
9. Assume that 1
x1 < x2 . We have
jf (x2 )
f (x1 )j =
1
x22
1
x21
=
=
=
Now
since x1
Thus
x21 x22
x22 x21
jx1 + x2 j jx1 x2 j
x21 x22
x1 + x2
(x2 x1 )
x21 x22
x1 + x
Math 534A Fall 2013
Problem Set # 6
Solutions
1. We have
0 () =
2
2
= 2
Note that
2
lim | 0 ()| = lim 2 = 0
+
Therefore, there exists 0 such that
| 0 ()| 1 if |
Since | 0 ()| is continuous, there exists 0 such that
| 0 ()| if |
Therefore
| 0 ()| 0 =
Math 534A Fall 2013
Problem Set # 5
Due on Thursday, October 31
In problems 1-3,
a) Determine the derivative function 0 directly from the definition of 0 (you need to
specify the domain of 0 ),
b) Determine the dierential of ,
c) Determine such that
( +
Math 534A Fall 2013
Problem Set # 8
(not to be turned in)
In problems 1-9 make use of a comparison test in order to determine whether the improper
integral converges or diverges. Do not try to evaluate the integral (in the case of an integral on
a bounded
Math 534A Fall 2013
Problem Set # 4
Due on Tuesday, October 22
1. Prove that
3 22 2 3
= 13
3
( 3)
lim
in accordance with the - definition of the limit.
Hint: Divide and then set = 3 + .
2. Show that
1
13 2
=
8 8
12
by finding a function that is continuous
Math 534A Fall 2013
Problem Set # 7
Solutions
1. Lets set
() =
Z
()
so that () = ( (). By the chain rule and the Fundamental Theorem of Calculus,
!
()
()
() =
=()
= ( ( ()
()
2 a) By the Fundamental Theorem of Calculus,
Z
2
2
2
2
=
0
2 b) By
Math534A - Fall 2016
Homework Assignment 6
Jrme Gilles
Exercise 1
Use the formal definition to prove that
lim
n
n
1
+
2
n
= +.
Exercise 2
Use the formal definition to prove that
lim n2 + n = .
n
Exercise 3
Show that the sequence cfw_an =
n
2n
3n+1
o
is i
Math534A - Fall 2016
Solutions Homework 1
Jrme Gilles
1
Exercise 1
1. x C, P (x, U ),
2. x U, P (x, C),
3. x C, y U, Q(x, y).
2
Exercise 2
1. f (x) 2 xn 1,
2. x 6= 13 3x + 1 6= 0,
3. b a (x > a) (x b).
3
Exercise 3
1. > 0, N N, n N, |N an 1| ,
2. > 0, > 0
Math534A - Fall 2016
Solutions Homework 9
Jrme Gilles
Exercise 1
We want to prove
n N, x R, > 0, > 0, h, |h| < |fn (x + h) fn (x)| < .
If n = 1 then f1 (x) = x thus |f1 (x + h) f1 (x)| = |x + h x| = |h|. In order to have |f1 (x + h) f1 (x)| < ,
we can cho
Math534A - Fall 2016
Homework Assignment 10
Jrme Gilles
Exercise 1
Let c R a constant and a function f (x) differentiable at x. Prove that (cf )0 (x) = cf 0 (x).
Exercise 2
Let f (x) and g(x) two function differentiable at x. Prove that (f + g)0 (x) = f 0
Math534A - Fall 2016
Solutions Homework 11
Jrme Gilles
Exercise 1
If we denote f 1 (x) = ln x, we have
d
d 1
(ln x) =
f (x).
dx
dx
But we have f (y) = ey thus
df (y)
dy
= ey and
df (y)
= eln x = x.
dy y=f 1 (x)=ln x
Therefore, using the given formula, we
Math534A - Fall 2016
Homework Assignment 11
Jrme Gilles
Exercise 1
Use the formula of the derivative of an inverse function to prove that
d
dx (ln x)
= x1 .
Exercise 2
Give the differential of f (x) = (x2 + 3)ex and an approximation (without any calculato
Math534A - Fall 2015
Solutions Homework 3
Jrme Gilles
Exercise 1
We know that cfw_an is a converging sequence. Note that necessarily L 0. Then consider the two cases:
if L = 0, we have
> 0, N N, n N, n N, |an 0| < 2 .
but
Thus
|an 0| = |an | = | an |2
Math534A - Fall 2015
Homework Assignment 3
Jrme Gilles
Exercise 1
Let cfw_an a sequence
converging to L. We alsoassume that n N, an 0. Show that the sequence cfw_ an
).
converges to L. (hint: if L > 0 then an L = aan L
+ L
n
Exercise 2
Let cfw_an a se
Math534A - Fall 2016
Homework Assignment 5
Jrme Gilles
Exercise 1
Let S be the set of points defined by
S=
1
(1)n
/n N .
n
1.1. Show that sup S = 2
1.2. Show that inf S =
1
2
Exercise 2
Show whether the following sequences are increasing, decreasing or ne
Math534A - Fall 2016
Homework Assignment 2
Jrme Gilles
Exercise 1
Show that the triangle inequality becomes an equality if a and b are of the same sign.
Exercise 2
Show that
a > 0, x R, |x a| <
a
a
x> .
2
2
Exercise 3
Let Sn = 12 + 22 + 32 + . . . + n2 .
Math534A - Fall 2016
Homework Assignment 4
Jrme Gilles
Exercise 1
Use the formal definition of a Cauchy sequence to show that the sequence cfw_an
n=1 =
sequence.
1
n
n=1
is a Cauchy
Exercise 2
Use the formal definition of a Cauchy sequence to show tha
Math534A - Fall 2016
Homework Assignment 9
Jrme Gilles
Exercise 1
Let a function f defined by n N, x R, fn (x) = xn . Use the definition (the formulation with h is
easier to handle) to prove that fn is continuous over R.
Exercise 2
Let f and g two functio