MAT 319: Foundations of Analysis
Fall 2016

## Syllabus and Homeworks

Homeworks
Homework sets will be typically assigned weekly on Thursday and due next Th in class. They will be posted on this web site (NOT on blackboard site!). Late homework will not be accepted. However, grades for homework assignments may be dropped in cases of documented medical problems or similar difficulties.

Week Dates Notes Topics HW Due date
1 8/29 - 9/2 1-3: Introduction: what are real numbers? Integer numbers and induction; rational numbers and idea of a field; basic properties of real numbers. Read carefully section 3 of the textbook and subsections 4.1-4.4 of section 4.
In particular, work out ﻿﻿proofs of Theorems ﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿3.1, 3.2 and 3.5﻿﻿.﻿﻿﻿
Write down solutions for exercises 3.2, 3.4, 3.8, 4.6.
9/8
2 9/5 - 9/9 9/5, 9/6: No classes - Labor day 4: The completeness axiom. Archimedean property HW 2; solutions 9/15
3 9/12-9/16 5: Infinity, boundedness and unboundedness. 7: Sequences; limit of a sequence. 5.2, 5.6, 7.3 (a- j), 7.4, 8.1 9/22
4 9/19 - 9/23 8-9: Limit laws for sequences and formal proofs. 8.3, 8.7, 9.1, 9.4, 9.12*, 9.15, 9.16. Problem 9.12 is optional (worth extra credit); even if you do not solve it, you are allowed to use results of this problem to solve other problems.
5 9/26 - 9/30 Midterm 1 on Th, 9/29
6 10/3 - 10/7 This is the last week MAT 319/320 meet together 10: Monotone and Cauchy sequences; 11: Subsequences HW 5 for MAT 319 only 10/13
7 10/10 - 10/14 11. Subsequences continued; 12:lim inf and lim sup. 14: series HW 6 (for MAT 319 only): 11.4, 11.10, 12.4, 14.2, 14.6 10/20
8 10/17 - 10/21 14:series; 15: integral test and alternating series HW 7 for MAT 319 only postponed to 11/1
9 10/24 - 10/28 17, 18: continuous functions
10 10/31 - 11/4 Midterm 2: 11/3
11 11/7 - 11/11 20: limits of functions HW 8: 17.8, 17.10 (a), 17.12, 18.6, 18.10, 20.4, 20.8, 20.14, 20.20
12 11/14 - 11/18 28: derivatives
13 11/21 - 11/25 No classes on Nov 23-25: Thanksgiving break 29 Mean value theorem
14 11/28 - 12/2 32 The Riemann integral (up to Th. 32.5); 33 Properties of Reimann integral HW 9: 28.6; 28.8, 29.2, 29.12, 29.14, 32.2, 32.6, 33.4 12/8
15 12/5 - 12/9
12/16 Final exam