Math 117 Fall 2014 Lecture 8
(Sept. 15, 2014)
Reading: 314 Notes: Sets and Functions 1; Bowman 1.A.
Sets.
Definition 1. A set is a collection of objects. Each object is called a member (or element)
of this set.
Notation. We use a 2 A to mean a is a member
Math 117 Fall 2014 Homework 4 Solutions
Due Thursday Oct. 9 3pm in Assignment Box
Question 1. (10 pts) Prove the following statements by denition.
n!
a) (2 pts) limn!1 nn = 0.
h
1
b) (2 pts) limn!1 p 2
+p
n +1
1
n2 + 2
+ + p
1
n2 + n
2
c) (2 pts) The seq
Math 117 Fall 2014 Lecture 23
(Oct. 15, 2014)
Reading:
Let fan g be a sequence. Recall:
If fan g converges, then fan g is bounded. Note that limn!1 an = 1 are called fan g
diverges to +1/1.
If fan g is bounded, then it has a convergent subsequence.
The se
Math 117 Fall 2014 Lecture 19
(Oct. 6, 2014)
Reading:
In the following a; b; L; M 2 R. The cases of one or more of them are +1 or 1 are left
as exercises. Please make sure you work on these cases some of them may not be that
straightforward and compare yo
Math 117 Fall 2014 Lecture 21
(Oct. 9, 2014)
Reading: Bowman 2.E.
Note that convergence means converging to a number. It does not include the cases an !
+1 and an ! 1. Please make sure you think about whether the following results still
apply to these two
Math 117 Fall 2014 Midterm 2 Review Problems
Midterm 2 coverage:
Lectures 12 - 25 and the exercises therein.
Required sections in Dr. Bowman's book and my 314 notes.
Homeworks 3 - 5.
The exercises below are to help you on the concepts and techniques. The
Math 117 Fall 2014 Lecture 18
(Oct. 3, 2014)
Reading:
Some leftovers.
a; L 2 R. Denition for limx!af (x) = L is not true.
9" > 0 8 > 0 9x0 < jx aj < ;
jf (x) Lj > ":
(1)
Remark 1. Note the dierence between the above statement and limx!af(x) = L,
/
which m
Math 117 Fall 2014 Lecture 24
(Oct. 16, 2014)
Reading:
Let fan g be a bounded sequence. Recall:
Can dene the set of accumulation points:
A(fan g) := fa 2 Rj 9ank ! ag:
Have seen:
p
A
n 2
= [0; 1]:
A(f(1)n g) = f1; 1g;
(1)
(2)
p
Exercise 1. Let an =
Math 117 Fall 2014 Lecture 20
(Oct. 8, 2014)
Reading:
In the following a; b; L; M 2 R. The cases of one or more of them are +1 or 1 are left as
exercises. Please make sure you work on these cases some of them may not be that
straightforward and compare yo
Math 117 Fall 2014 Lecture 25
(Oct. 17, 2014)
Reading:
Recall: For a bounded sequence fan g,
limsup an := lim sup ak ;
n!1 k>n
n!1
liminf an := lim
n!1
h
i
inf ak :
n!1 k>n
(1)
Theorem 1. Let fan g; fbn g be bounded sequences. Then
limsup (an + bn) 6 lims
Math 117 Fall 2014 Homework 2
Due Thursday Sept. 18 3pm in Assignment Box
Question 1. (5 pts)
a) (2 pts) Find two irrational numbers a; b such that both a + b and a b are rational.
b) (3 pts) Can you nd two irrational numbers a; b such that both a + b and
Math 117 Fall 2014 Homework 1 Solutions
Due Thursday Sept. 11 3pm in Assignment Box
Question 1. (5 pts) Prove that 11 is prime but 57 is not.
Proof.
11 is prime.
First for any number n > 11, nj 11. Now we check 1j 11; 11j 11 but
2j 11; 3j 11; 4j 11; 5j 11
Math 117 Fall 2014 Lecture 6
(Sept. 11, 2014)
What is ?
The ratio between the circumference and diameter of a circle; or the ratio between
the area and the square of the radius of a circle. But why are they the same number?
The usual high school proof rel
Math 117 Fall 2014 Lecture 9
(Sept. 17, 2014)
Reading: 314 Notes: Sets and Functions 2.1 (Open and Closed Sets: Optional); Bowman 1.G.
Operations on two sets (cont.)
Example 1. Prove that (A B) \ (B A) = ?.
Proof. Take an arbitrary x 2 (A B). By denition
Math 117 Fall 2014 Lecture 4
Reading:
(Sept. 8, 2014)
p
Required reading: Dr. Bowman's book 1.B.
Optional reading: 1.C.
p
2 is irrational, that is 2 is not rational.
p
Notation. The symbolic way of writing this is 2 2 Q. (Recall that 2 means belongs to)
/
Math 117 Fall 2014 Lecture 5
(Sept. 10, 2014)
Prehistory.
Before the invention of logarithm, people calculated multiplications through the following
trigonometric identities:
sin(A B) = sin A cos B cos A sin B;
cos(A B) = cos A cos B sin A sin B:
(1)
(2)
Math 117 Fall 2014 Lecture 7
(Sept. 12, 2014)
Reading: Bowman 1.H, 1.I, 1.J.
A bit more about numbers.
N: natural numbers; Z: integers; Q: rational numbers; R: real numbers.
Q is dense in R; Qc is dense in R.
Meaning: between any two real numbers there ar
Math 117 Fall 2014 Midterm Exam 1 Solutions
Sept. 26, 2014 10am - 10:50am. Total 20+2 Pts
NAME:
ID#:
There are ve questions.
Please write clearly and show enough work.
Question 1. (5 pts) Prove that
a) (2 pts) 19 is prime.
p
b) (3 pts) 19 is irrational.
Math 117 Fall 2014 Lecture 2
Number systems:
(Sept. 4, 2014)
N = f1; 2; 3; :g: Natural numbers;
Z = f: 2; 1; 0; 1; 2; :g: Integers;
Q: Rational numbers;
R: Real numbers;
C: Complex numbers;
And many more.
Relations between number systems: N Z Q R C.
A B f
Math 117 Fall 2014 Midterm 1 Review
Midterm 1 coverage:
Lectures 1 - 11 and the exercises therein.
Homeworks 1 & 2.
Required sections in Dr. Bowman's book and my 314 notes.
The exercises below are to help you on the concepts and techniques. The exam
probl
Math 117 Fall 2014 Lecture 3
(Sept. 5, 2014)
How detailed should your answers to Homework 1 be:
For example, to prove that the product of two odd numbers is still odd, you should write
something like:
Recall that a natural number a is odd if and only if a
We see that the old denition still applies.
A (c, d) where a (c, d).
1
Example 3. Study limx1 x3 .
1
Solution. Clearly the limit should be 1. The problem now is that f (x) = x3 is not
dened at x = 0. Lets see whether the old denition still applies.
1
Let