Homeworks, due Thursdays at 5:15pm,
in box outside my office (Olin 1269).
All HW's refer Rudin's Principles of Mathematical Analysis.
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- HW #0. Due Thursday Jan 24 at 5:15pm.
Read this
handout on good mathematical writing and turn in brief answers
to these questions.
Keep in mind the handout
and the homework
format as you write up your answers.
- Directly from the handout reading:
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1. What is a good rule of thumb for what you should assume of
your audience as you write your homework sets?
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2. Is blackboard writing formal or informal writing?
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3. Do you see why the proof by contradiction on page 3 is not really a
proof by contradiction?
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4. Name 3 things a lazy writer would do that a good writer wouldn't.
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5. What's the difference in meaning between these three phrases?
"Let A=12."
"So A=12."
"A=12."
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Now examine Section 1.1 of Rudin, showing that there is no rational p
that
satisfied p2=2.
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6. There are many places in his proof where he could have used symbols
to
express his ideas, but he does not. (e.g., "Let A be the set of
all
positive rationals p such that...") Why do you think he chooses
not to
use symbols?
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7. What would you change about his presentation if you were writing
for a
high school audience? Give a specific example.
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p.s. Be sure to subscribe to math-131-l (using listkeeper@hmc.edu) if
you are not already getting e-mails from the list.
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- HW #2. Due Feb 7.
A problem marked "R" means read, but do not do the problem.
Chapter 1 ( R3bcd, 4, 5, R7, 8, 9, 20 ) and
Problem S. For a real number a
and non-empty
subset of reals B, define: a + B =
{ a + b : b is in B }. Show that if B is bounded above, then sup( a + B ) = a + sup B.
[For 1.20, whenever the proof is
EXACTLY THE SAME as in Steps 3 and 4 of pp. 18-19, you do not need
to
re-write the proof. Just point out that the proof is the same as
in
Rudin's. But wherever the proof differs, BE SURE to POINT OUT HOW
IT DIFFERS, and VERIFY all new things.]
If you want to do rewrites,
see this guide.
p.s. Be sure to subscribe to math-131-l (using listkeeper@hmc.edu) if
you are not already getting e-mails from the list.
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Rewrites for HW#1 are also due.
If you want to do rewrites,
see this guide.
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- HW #4. Due Feb 21.
A problem marked "R" means read, but do not
do the problem.
Do Chapter 2 ( 2, R3, 4, 5, R6, R7, 8, 11[exclude d2] ).
Problem T. Prove that the Principle of Induction implies the
Well-Ordering Principle for N (the natural numbers).
[Hint: Let
L(n)
be the statement "if A is a subset of N that contains
a number <= n, then A has a least element".
Now prove L(n) holds for all n by induction.]
Rewrites for HW#2 are also due.
If you want to do rewrites,
see this
guide. Save a copy since you probably won't get the homework
back in time to study for the exam.
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- EXAM 1. Available Monday Feb 25 (afternoon).
Though test is due FRIDAY Mar 1 at 2:30pm.
Rewrites for HW#3 are still due Thursday 2/28.
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- HW #5. Due Mar 7. A problem marked "R" means read, but do not
do the problem. There are no re-writes on this homework and forward.
Do Chapter 2 ( 9ab, 9cd, 9ef, R10, 12, R14, 16, 22, R23 ).
In Problem 14, give an example of a cover that is not a
nested collection of open sets.
Rewrites for HW#4 are also due.
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No further rewrites accepted.
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Spring Break Week
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- HW #7. Due Mar 28.
A problem marked "R" means read, but do not do the problem.
Do Chapter 2 ( 20, R26 ) Chapter 3 ( 1, 3, R4, R16, 20, 23, R24, R25 )
Hint on 3.3: can you show the sequence is increasing? Induction
may be of help here.
Please have a look at these problems before spring break. Have a great break!
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- HW #8. Due Thu Apr 11.
A problem marked "R" means read, but do not do the problem.
Do Chapter 3 ( 4, 7, 8, R9, 10, 16a, 16bc, R17 )
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- HW #9. Due Thur Apr 18. A problem marked "R" means read, but do not
do the problem.
Chapter 4 ( 1, 2, 3, 4, R6, 8 )
and this
Problem M. Carefully write out the details to the proof of Theorem
4.10(part a).
On the reading problem 4.6,
you may assume that E is a subset of real numbers and
f is a real-valued function, and the distance in R2 (where the graph
lives) is the usual Euclidean metric.
Don't forget the Moody Lecture
on Friday April 19 at 7pm: "The Mathematics of Doodling". It's
related to analysis, but will be accessible to all! Invite your
friends! Watch your e-mail for how to earn extra credit.
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- HW #10. Due Apr 25.
A problem marked "R" means read, but do not do the problem.
Do Chapter 4 ( 9*, 10, 11**, 12, 14, R16, 18, R19 ).
*The definition of diameter can be found on page 52.
**You do NOT have do the 2nd part of Problem 4.11, where it says "Use
this result to give..."
Please come see me or the tutors to discuss these problems!
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- HW #11. Due May 2.
Do Problem E. Use the mean value theorem to show that e^x is greater
than or equal to (1+x) for all x in R. (You may assume knowledge of
the derivative of e^x.)
and Chapter 5 ( 1, 4, 13ab*, R13cdefg, R25abd, 25c )
and Chapter 7 ( 1, R2, R3 ).
*In Problem 5.13, there is a typo (in some editions of the book): the
xa should say |x|a.
The Final Exam will NOT be a take-home. See details in your e-mail.
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