Math 3333 --Homework so far

Assigned Problems (the text contains many more problems that you could or should try as exercises, for example the

DATES BELOW ARE APPROXIMATE AND MAY CHANGE, SEE FLASHING RED LIGHT ON MAIN PAGE FOR ACCURATE INFORMATION.
Homework 0: Homework for Chapter 1 of text - 5th Edition numbers: 1.1 Ex 2, 1.2 Ex 3, 11, 13, 15, 17, 18, 22. 1.3 Ex 2, 3, 6a-f, 7. 1.4 Ex 1, 9, 11, 19c, 23, 24. The following are 4th (3rd) Edition numbers for the same exercises (numbers in parentheses refer to the equivalent problems in the third edition of the text--if you have the 3rd edition of the text you would do all the 4th edition numbers, except where parentheses indicate that the 3rd Ed is different, in which case do the number in parentheses): Exercises 1.2, 2.3, 2.5 (not f), 2.11 (2.7), 2.13 (2.9), 2.15 (2.11), 2.16 (2.12), 2.20 (2.16), 3.2, 3.3, 3.6a-f, 3.7 (3.8), 4.1, 4.3, 4.5, 4.13c (4.7), 4.17 (4.11), 4.18 (4.12). You do not need to do them all. They will be collected on Friday August 25, but the grade will be essentially just for turning something in. Instead, there will be a 10 minute quiz on that day containing some of these problems.
PLEASE INDICATE WHAT EDITION OF THE TEXT YOU ARE USING ON THE FIRST PAGE OF EVERY HOMEWORK ASSIGNMENT

Homework 1 (due Monday August 28): Text 5th Ed Section 3.1 numbers 3, 7, 17a,b, 23, [4th Ed (and 3rd Ed) numbers 10.4, 10.7, 10.15a,b (10.16a,b), 10.21 (10.13).]

  • Homework 1 selected solutions (and point allocation).

    • Homework 2 (due dates may be found by the `red flashing dot' on the main page). - PLEASE INDICATE WHAT EDITION OF THE TEXT YOU ARE USING ON THE FIRST PAGE OF EVERY HOMEWORK ASSIGNMENT: Text 5th Ed {Section 3.2 number 3a,b,c, Section 3.3 numbers 1a-c, 5, 6, 8}, {Section 3.3 numbers 3degi (prove these), 4e,i (prove these), 7, and A, B below}. 4th (and 3rd) Ed numbers: {11.3a,b,c, 12.1a-c, 12.5, 12.6, 12.8,} {12.3d,e,g,i (prove these), 12.4e,i (prove these), 12.7, and A, B below}]

    Also: (A) Show that every nonempty subset of R which is bounded below has a greatest lower bound or inf (this falls out of the proof of 4th (and 3rd) Ed Question 12.7 (b) (Section 3.3, 7 (b) in 5th Ed) with k = -1). (B) NOTE SLIGHT CHANGE: Do case (1) (that is [a,b]) and (5) of the last theorem in Section 3.3 of classnotes (page 21 in class notes), in both directions (($\Rightarrow$) and ($\Leftarrow$)).

  • Homework 2 solution (and point allocation).

    • Homework 3 (due dates may be found by the `red flashing dot' on the main page). PLEASE INDICATE WHAT EDITION OF THE TEXT YOU ARE USING ON THE FIRST PAGE OF EVERY HOMEWORK ASSIGNMENT. Text 5th Ed Section 3.4 numbers 2ab, 3a-c (prove these), 4a-c (prove these, and also find the accumulation points of these sets, with proofs), 5a-c, 11, 13 [4th (and 3rd) Ed numbers: Numbers in parentheses refer to the equivalent problems in the third edition of the text: 13.2ab, 13.3a-c (prove it), 13.4a-c (prove it, and also find the accumulation points of these sets, with proofs), 13.5a-c (prove it), 13.11 (13.7), 13.13 (13.9).]

    Also answer: (A) Which of the 9 types of intervals are open, and which are closed? (B) Give an example to show that a union of closed sets need not be closed. (C) Show that the closure of the rationals is the reals, and that the closure of the irrationals is the reals. (D) Show that the interior of a set is open. Also show that Int(S) is the largest open set contained in S. (E) Show that the closure of a set S is the smallest closed set containing S.

  • Homework 3 and 4 selected solutions (and point allocation).

    • The key to the last couple of questions on HW 3 will be the first couple of questions on the key to HW 4
    Homework 4. The due dates may be found by the `red flashing dot' on the main page. See latest version of course notes for a prettier version of the following. (1) 3.5 Exercise 1 from the textbook 5th Ed (Exercise 14.1 in 3rd or 4th Edition). (2) Explain why each of the following sets are not compact: (a) \ [1,3), (b) \ the set of natural numbers N, (c) \ the set { 1/n : n \in N }, (d) the set of rational numbers in the interval [0,2]. (3) Which of the 9 types of intervals are compact? (4) Show that any isolated point of S is in Bdy(S). (5) \ Show that the closure of a bounded set is bounded. (6) Read the proofs in the Section on compact sets which we skipped in class.

  • Homework 3 and 4 selected solutions (and point allocation).

    • Homework 5. (Approximately two are due each class day as marked by curly braces below; the due dates may be found by the `red flashing dot' on the main page (Upcoming due dates watch this space if we need to adjust these dates)). 5th Ed numbers: Section 4.1: {2, 6c-e, 7b,c}, {15a, Section 4.2 5b,f,i,l, 6 a,b, 7, 14}. [3rd and 4th Ed numbers: {16.2, 16.6c-e (16.4c-e), 16.7b,c(16.5b,c),} {16.15a (16.13a), 17.5b,f,i,l, 17.6 a,b, 17.7, 17.14}.]

  • Homework 5 selected solutions (and point allocation).

    • Homework 6. the due dates may be found by the `red flashing dot' on the main page. 5th Ed numbers: Section 4.3: {1, 2, 3a,d, 4a-c,} {15. Section 4.4: 2, 4} (just compute liminf and limsup) [3rd and 4th Ed numbers, numbers in parentheses refer to the equivalent problems in the third edition of the text: {18.1, 18.2, 18.3a,d, 18.4a-c,} {18.15 (18.11), 19.2ac, 19.4 (just compute liminf and limsup).}
    Read the parts we skipped above on the liminf and limsup. For those interested, read some of the proofs of parts of Theorem 4.14 on the limsup and liminf.

  • Homework 6 selected solutions (and point allocation).

    • Homework 7--the due dates may be found by the `red flashing dot' on the main page. 5th Ed numbers: Section 5.1 Q 6, 7, 14; 5.2 Q 1, 3, 9, 18. [3rd and 4th Ed numbers, numbers in parentheses refer to the equivalent problems in the third edition of the text: 20.6 (20.4), 20.7 (20.5), 20.14 (20.12), 21.1, 21.3, 21.9, 21.18 (21.16).

    Note the list of homework in the typed notes may possibly be wrong: go by the website. Also for Homework 7: A) Prove from the definition that the limit as x approaches -1 of (2 x^2 + 11x + 17)/(x+3) is 4. B) Complete the proof of Proposition 5.4. C) prove from the definition that x^3 is continuous at x = 2.

  • Homework 7 selected solutions (and point allocation).

    • Homework 8--the due dates may be found by the `red flashing dot' on the main page: [3rd and 4th Ed numbers: 22.1(c), 22.2, 22.4, 22.7.] 5th Ed. numbers: 5.3: 1(c),2 4,7.

    See course notes for an additional 3 problems not in the text.

  • Homework 8 selected solutions (and point allocation).

    • Homework 9--the due dates may be found by the `red flashing dot' on the main page: [3rd and 4th Ed numbers, numbers in parentheses refer to the equivalent problems in the third edition of the text: 25.1, 25.6, 25.7a,d, 26.5 d,j, 26.17 (26.15).] 5th Ed. numbers: 6.1: 1, 6, 7a,d; 6.2: 5d,j, 17.

    See course notes for a prettier version of the following--Also: Use the MVT to show that 1+nx is less than or equal to (1+x)^n for all natural numbers n and for all x> 0 (this is called *Bernoulli's inequality*. Also prove or look up and read carefully the proof of the Intermediate Value for Derivatives: Let f be differentiable on [a,b] and let k be a real number between f'(a) and f'(b). (these are the derivatives of f at a and at b). Then there exists c in (a,b) s.t. f'(c)=k.

  • Homework 9 selected solutions (and point allocation).

    • Homework 10: --the due dates may be found by the `red flashing dot' on the main page. [3rd and 4th Ed numbers, numbers in parentheses refer to the equivalent problems in the third edition of the text: 29.1, 29.2, 29.7, 29.15(29.13), 30.5, 30.9 (Hint: do Exercise 29.17(29.15) first), 30.10. Much harder, hence optional: 29.18(29.16).] 5th Ed. numbers: 7.1: 1, 2, 7, 15; 7.2: 5, 9 (Hint: do Exercise 7.1 (17) first), 10. Much harder, hence optional: 7.1:18.

    See course notes for a prettier version of the following--Also: (1) prove that 1/x is not uniformly continuous on (0,1]. (2) Prove that f is Riemann integrable on [a,b] iff there exists a number A such that for every epsilon > 0, there exists a partition Q of [a,b] such that | R(f,P) - A | < \epsilon for any partition P which is finer than Q, and for any Riemann sum R(f,P) corresponding to partition P.

  • Homework 10 selected solutions (and point allocation). Note on Q 29.7 the point allocation on the key is wrong, most of the points should be for the first two equalities.