Math 478: HW 9
Due Thursday, November 13th, 2014
Problem 9.1:
Let cfw_xn , cfw_yn be two convergent sequences such that xn yn for all n. Show
lim xn lim yn .
n
n
Problem 9.2:
a) Prove the following
inf xn lim inf xn lim sup xn sup xn .
n
n
n
n
b) Show li
Binghamton University
EXAM 1: MATH 478
Real Analysis
FALL 2014
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Math 478: Real Analysis 1
Study Hints
Some advice
This is a serious and dicult course. To do well, you must work hard, and you are ultimately responsible for your success in the course. Your main objective should be to learn and understand the concepts
an
1
Lecture 9. More Limits (Section 4.2)
- Denition of one-sided limits
(a) Let
f (x) =
1
By choosing two sequences an = n
1
n
f (bn ) = e
ex , x 0,
0,
x < 0.
and bn =
1
n
, it follows that f (an ) = 0 and
1. Therefore, limx0 f (x) does not exist. On the o
1
Lecture 10. Continuity (Section 4.3)
Continuity
(a) Denition:
In the denition of limxc f (x) = L, if we replace 0 < |x c| < by |x c| < ,
where we assume that f is dened at c, then we have |f (c) L| < . Since > 0
1
is arbitrary, from choosing, say, = 2
1
Lecture 5. Innite Series (Section 3.5)
nth term test
n2
n=1 n2 +3 .
(a) Consider the series
It is easy to see that the nth term an =
n2
n2 +3
converges
to 1 as n . Therefore, there exists some N > 0 such that n > N implies that
1
an > 2 . This implies t
1
Lecture 8. Limits of Functions (Section 4.1)
Disclaimer
We will not specify explicitly the domain D of a function f , that is, when we say a
function f (x), x is always in D, so we omit the assumption x D. For example, the
statement 0 < |x c| < implies
1
Lecture 11. Topology in R (Section 4.4)
Disclaimer
In this course the topology in R is induced by the standard metric d(x, y) = |x y|. So
an open interval is open, a closed interval is closed, etc.
Bolzano-Weierstrass Theorem
If cfw_an R is a bounded s
Math 478: HW 8
Due Thursday, November 6th, 2014
Problem 8.1:
Use that R is complete to show that Rk is complete.
Problem 8.2:
Let E be a closed subset in a complete metric space X. Show E is a complete metric space.
Problem 8.3:
Let cfw_xn be a sequence
Math 478: HW 11
Due Thursday, December 4th, 2014
Problem 11.1
Let (X, dX ), (Y, dY ) be two metric spaces, E X and p E a limit point of E. Show f is continuous
at p if and only if for every sequence cfw_xn E such that xn p we have f (xn ) f (p). (Do not
Math 478: HW 10
Due Wednesday, November 26th, 2014
Problem 10.1:
Show directly from the denition that the following functions are continuous at a given point x0 :
f : R R, where f (x) = x2 , and x0 = 2
1
f : R2 R, where f (x, y) = (xy) 3 , and x0 = (0,
Math 478: HW 12
Due Thursday, December 11th, 2014
Problem 12.1
Let f (x) = x4 x2 11x + 10 and E = [, 2014]. Answer the following questions about f . Do not do any
computations (except for plugging in two points in part 5). Justify your answers by writing
Basic Topology Denitions
Denition 1. Let X be a metric space. All points and sets mentioned below are elements and subsets of
X.
A neighborhood of p is a set Nr (p) = cfw_
the radius of Nr (p).
. The number r is called
A point p is a limit point of the
Math 478: HW 7
Due Friday, October 31st, 2014
Problem 7.1:
Let xn = (1)n . Prove that there is no real number x0 such that xn x0 as n .
Problem 7.2:
Let cfw_xn be a sequence in a metric space X.
a) Suppose cfw_xn has a nite range. If xn x0 , is x0 a lim
Math 478: HW 3
Due Wednesday, 9/24/2014
p. 23: 13 (instead of x, y being complex suppose x, y Rk , k > 0.)
Problem 1: Show an inner product (, ) on a vector space H, induces a norm on H. (Recall we can
1
dene |x| = (x, x) 2 . Show | | satises the denition
Math 478: HW 6
Due Thursday, October 23rd, 2014
Problem 6.1:
a) Show directly from the denition (without using Heine-Borel Thm) that an open interval (0, 1) is not
compact.
b) Show directly from the denition (without using Heine-Borel Thm) that R is not c
Math 478: HW 4
Due October 2nd, 2014
Problem 1:
Complete and then prove the following theorems about cardinalities of unions of sets (do a direct proof;
do not say it follows by a theorem/corollary from class/book):
Theorem 0.1. Let n N, and E1 , E2 , En
Math 478: HW 2
Due September 19th, 2014
Problem 1: Show that if
> 0, there exists a natural number N such that
1
< .
N
From the book: p. 22: 5, 6 a, b, c; 8;
1
1
Lecture 7. Tannerys Theorem (Section 3.7)
Tannerys Theorem
(a) A heuristic proof of the formula
1
1 + 2n
1 + 2n
1 + 2n
1 + 2n
= lim
+
:
+
+ + n n
2 n 2n 3 + 4 2n 32 + 42 2n 33 + 43
2 3 + 4n
Step 1. Write down the general term for the above finite sum: F