Homework
Math 554.001⊕
Spring 2025
Prof. Girardi

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Textbook: Introduction to Real Analysis, 4th ed., by Bartle and Sherbert.

HW set section
pages 		note to yourself of
due date 		Homework Instructions⊕     and     Handouts⊕
1 1.2
15-16 		Th 1/30 Since induction (§1.2) is covered in Math 300, there will be no formal lectures over §1.2.
Here is a summary and 3 examples of proofs by induction.
  1. ER 0.0.1. Read the Homework Instructions (posted above). Write at least 5 items you want to remember.
  2. ER 0.0.2. From the class Handout page (see above link), read all handouts under Writing Guidelines (WG). Write at least 5 items you want to remember.
  3. ER 1.2.5
  4. ER 1.2.14   (Use induction. Do not use calculus.)
  5. ER 1.2.15   (Use induction. Do not use calculus.)
  6. ER 1.2.20
2 1.1
10-11 		Th 2/6
  1. ER 1.1.1
  2. Verify Thm 1.1.4+ part (2), which is a DeMorgan Law. You may use symbolic logic language, as we did in class for part (1). Note this problem is a variant of ER 1.1.4.
  3. ER 1.1.5.
  4. ER 1.1.7
  5. ER 1.1.10
  6. ER 1.1.19a. You may use (3) and (4) from Functions and Sets.
  7. ER 1.1.19b. You may use (3) and (4) from Functions and Sets.
3 1.1
10-11

2.1
30-31

Th 2/13
  1. Verify set equality (8) from Functions and Sets, which is about preimages of unions. You may use symbolic logic language, as we did in class for (9). This is a variant of ER 1.1.15.
  2. ER 1.1.21. You may use the result shown in class that if f and g are injective then g ∘ f is injective.
  3. ER 1.1.23
  4. ER 2.1.17. Will use this ER and Thm 2.1.9 lots.
  5. ER 2.1.18
  6. ER 2.1.22 (a) and (b). For (a), do by using Bernoulli's inequality (Ex. 2.1.13).
  7. ER 2.1.51 (click the link). Not in book. An application of the Geometric-Arithmetic Mean inequality (see book's Example 2.1.13).
4 2.2
35-36 		Th 2/20
  1. ER 2.2.51 (click the link). Not in book. The 4 triangle inequalities.
  2. Variant of ER 2.2.18a (click the link) max/min via midpt
  3. Variant of ER 2.2.16 (click the link) ε-NBHDs: ∩ and ∪
  4. Variant of ER 2.2.17 (click the link) making disjoint ε-NBHDs
5 2.3
39-40 		Th 2/20
  1. Variant of ER 2.3.5 (click the link) max/min and sup/inf chart
  2. Variant of ER 2.3.10 (click the link) sup of a union
  3. Variant of ER 2.3.11 (click the link) sup/inf for A⊆B
6 2.4
44-46 		Th 2/27
  1. Variant of ER 2.4.1 (click the link) sup for 1-1/n
  2. Variant of ER 2.4.2 (click the link) sup and inf for 1/j - 1/k
  3. Variant of ER 2.4.4a (click the link) inf when multiply a set by postive number
  4. Variant of ER 2.4.4b (click the link) sup/inf when multiply a set by negative number
  5. Variant of ER 2.4.8 (click the link) sup of the range of sum of 2 function

Findable from URL: http://people.math.sc.edu/girardi/w554.html