### MATH 401/501 - HOMEWORK PROBLEMS - Fall 2012

Numbered exercises are from Tao's Book. For this problems you can only assume results and exercises prior to the given exercise (if you are going to use a previous result, make sure you state it, of course if it has not been proved in class and is not one of the other homework problems, you can attempt to prove those intermediate results, but this is not required.

Homework 1 (due Tu Aug 28, 2012):
For the first two problems assume arithmetic and whatever you know about inequalities:

1. Show by induction that for all n natural numbers 1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.
2. Show by induction that n^2 <= 2^n for all natural numbers n>3.
3. 2.3.1 (p.32 2nd ed, p.36 1st ed) (multiplication is commutative)

Homework 2 (due Th Sep 4, 2012):

1. 2.2.3 (d) OR (f) [p.29 2nd ed, p.33 1st ed] (Properties of order)
2. 2.3.5 [p.32 2nd ed, p.36 1st ed] (Euclidean Algorithm). Also show that the m,r you find are unique.

Homework 3 (due Th Sep 11, 2012):

1. Section 3.1: 3.1.10 [p.46 in 2nd ed.]
2. Section 3.3: 3.3.2, 3.3.5 (injective only, in class we did surjective), or 3.3.7 (you can use results stated in previous exercises, eg. 3.3.2) [do at least one of the three exercises proposed] [p 55-56 2nd ed.]
3. 3.6.5 and 3.6.9 [p.72 2nd ed.]
4. Bonus: exercise 3.6.6 in p. 72, 2nd ed. (the bonus you turn in to me, separated from the rest of the homework).

Homework 4 (due Th Sep 18, 2012):

1. Section 4.1: 4.1.2 (negation in Z is well defined), 4.1.5 (integers have no zero divisors).
2. Section 4.2: 4.2.6 (multiplication by a negative rational reverses order of inequality).
3. Section 4.3: 4.3.2 (f) (prove part (f) of Prop 4.3.7 about epsilon-close).
4. Bonus: in Section 4.4 Exercise 4.4.1 (prove the interspersing of integers by rationals).

Homework 5 (due Tu Oct 9, 2012):

1. Section 5.2: 5.2.1
2. Section 5.3: 5.3.5 and 5.3.2 (only the second part, I showed in class that xy is a real number, but not that it was well defined).
3. Section 5.4: 5.4.4 (I proved it today in class).
4. Bonus 5.4.3 and 5.4.5.

Homework 6 (due Th October 18, 2012):

1. Section 5.5: exercise 5.5.1 (hint: use a mirror)
2. Section 6.1: exercises 6.1.3, 6.1.5 and 6.1.8(a)(e)
3. Bonus 6.1.8(g)
4. Reading assignment: Section 6.2 (p.132-135)

Homework 7 (due Th October 25, 2012):

1. Section 6.3: exercise 6.3.4
2. Section 6.4: exercises 6.4.5 (squeeze test)
3. Section 6.5: exercise 6.5.2 (use ex 6.3.4) (some specific limits)
4. Section 6.6: exercises 6.6.2 and 6.6.4. (about subsequences)

Take home test (due Tu Nov 6, 2012):
Read Sections 7.2 and 7.3 about series (use any result from 7.1)

1. Section 7.2: exercises 7.2.2 (Cauchy test for series), 7.2.3 (zero or divergence test), 7.2.4 (absolute convergence), 7.2.6 (telescoping series).
2. Section 7.3: exercise 7.3.1 (comparison test).

Homework 8 (due Th November 13, 2012):

1. Section 9.1: Exercise 9.1.15 (adherent points and sup/inf)
2. Section 9.1: Exercise 9.3.3. (limits are local)
3. Section 9.1: Exercise 9.4.5. (composition preserves continuity)
4. Section 9.1: Exercise 9.6.1. (some examples of continuous functions)
5. Bonus Exercise 9.3.4.

Homework 9 (due Tu after Thanks Giving):

1. Exercise 9.7.2 (fix point)
2. Exercise 9.8.1 and 9.8.2 ( more examples for you to create)
3. Exercise 9.9.2 (uniform continuity iff equivalent sequences are mapped into equivalent sequences)
4. Show that the function square root of x is uniformly continuous on [0,oo). Show first that is uniformly continuous on [0,1], then show is uniformly continuous on [1,oo) and finally make sure things are Ok at x=1.

Homework 10 (due Tu Dec 4):

1. Section 10.1: Exercise 10.1.13(g) (prove the "baby quotient rule" or "reciprocal rule")
2. Section 10.2: Exercises 10.2.2 and 10.2.3 (examples) Exercise 10.2.5 (deduce Mean value Theorem from Rolle's Theorem)
3. Section 10.3: Exercises 10.3.2 and 10.3.3 (examples)
4. Section 10.4: Exercise 10.4.1

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Last updated: November 15, 2012