Math 131A/2 Winter 2002 Handout #1a Instructor: E. Eros, MS 6931. Lecture Meeting Time: MWF 2:00PM-2:50PM Location: MS 5117 Recitation TA: to be announced, MS 5117 T 2:00P-2:50P Oce hours (tentative):
Elementary Analysis
Kenneth A. Ross
Selected Solutions
Angelo Christopher Limnios
EXERCISE 1.2
Claim:
P (n) = 3 + 11 + + (8n
Proof :
5) = 4n2
n 8n 2 N
By induction. Let n = 1. Then 3 = 4(1)2 (1) = 3,
Thus Ak g BnVnJc (Why?). So rst taking the supremum over 19 in this
inequality, lim inf an s Bme and then taking the infemum over In gives
lim inf an 5 lim sup an. 0
If c is a limit point of the seque
a statement is true; it suices to show that it is not false; to show that
a statement is false, it suices to show that it is not true; this is the
principle underlying the powerful technique of proof
1. SOLUTIONS TO HW5
Problem #4 parts d and e
o Ifq>0thenm>yiffmq>yq.
If q > 0 then q = % m,n > 0. We have from Problem 1 that 20% > 31%.
Also, we know that if z and w are real numbers and n > 0 is an
Supplemental handout - The basics of mathematical logic
0 The purpose of this supplemental handout is to give a quick introduc-
tion to mathematical logic, which is the language one uses to conduct
ri
3 + 3 : 5 is not. Another example: 2 + 2 = 4 and 2 + 2 = 4 is
true, even if it is a bit redundant; logic is concerned with truth, not
efciency.
Due to the expressiveness of the English language, one c
Logic is a skill that needs to be learnt like any other, but this skill
is also innate to all of you - indeed, you probably use the laws of
logic unconsciously in your everyday speech and in your own
1. SOLUTIONS TO HW3
Problem #8
There are at least two ways to do this problem; a direct induction or the more
elegant use of the Euclidean algorithm. We present the latter. We wish to exhibit
a biject
Math 131A: Preparation Questions for Quiz 2
1. For each of the following sequences (sn )
n=1 prove that (sn ) converges. You should use the
definition of convergence given in lectures (which, at the m
Math131a
R. Kozhan
Midterm2 Summary
Midterm2 will be focused on the sections listed below, and will not explicitly test the
knowledge of the material included for Midterm1. However the student is assu
Selected answers HW #6
Problem 15.2
Determine which of the following series converge. Justify P
your answers.
P
(a) [sin(n/6)]n
(b) [sin(n/7)]n .
answer:
(a) does not converge, because if we assume (a
Math131a/3, R.Kozhan
Midterm 2 solutions
Nov 21, 2011
Name:
UID:
Instructions:
If you get stuck, move on to the next question! You dont have a lot of time.
Show all work if you want to get full cred
Selected answers HW #8
is uniformly continuous on [0, ).
(b) use (a) and exercise 19.6(b) to prove that x is
uniformly continuous on [0, ).
proof:
Problem 18.5
(a) f is continuous on [0, ) implies it
Math131a/3, R.Kozhan
Midterm 1
Oct 24, 2011
Name:
UID:
Instructions:
If you get stuck, move on to the next question! You dont have a lot of time.
Make sure to look at the last question. Its short an
Selected answers HW #7
Problem 17.10
Prove that the following functions are discontinuous at
the indicated points.
(a) f (x) = 1 for x > 0 and f (x) = 0 for x 0, x0 = 0.
Problem 20.10
(b) g(x) = sin(1
Math131a/3
Midterm 2
Nov 19, 2010
Name:
Instructions:
There are 4 problems, 10 points each.
No books, notes, electronics (incl. calculators and cell-phones) are allowed.
Do not use your own scratch
Math131a/3
Midterm 1
Oct 20, 2010
Name:
Instructions:
There are 4 problems, 10 points each.
No books, notes, electronics (incl. calculators and cell-phones) are allowed.
Do not use your own scratch
are considered to be neither true or false (in fact, they are usually not
considered statements at all). A more subtle example of an ill-formed
statement is
0/0 = 1;
division by zero is undened, and s
or efciency; the reason is that truth is objective (everybody can agree
on it) and we can deduce true statements from precise rules, whereas
usefulness and efciency are to some extent matters of opini