Prof. Thea Pignataro
Recommended higher level reading: Introductory Real Analysis, Kolmogorov & Fomin
Also recommended any Dover Books (good classic texts)
Review Chapter 1 as MATH 30800
Remaining notes by prof. online: http://1drv.ms/1KlFihR
It seems that most parts of advanced calculus is about proving things taken for granted in previous calculus classes. There should be indication of this in the beginning of calculus class, so I don't have to waste my time figuring out those proofs myself back then.
Sites with Textbook solutions:
https://www.math.ku.edu/~slshao/spring2015math500.html
http://www.math.ucla.edu/~hendricks/Math131AF
Chapter 2
Section 7
Definition:
A sequence is a function whose domain is of the form {n in Z: n >= m for some m in Z} pg. 33
Denoted S(n) = (S_n) = (S_1,S_2,S_3,...)
S_n in R (real).
set of integer Z comes from german Zahlen.
A sequence is always infinite. However, its values may not be infinite.
Definition:
S_n converges to S in R (called the limit) iff for all e > 0,...pg. 35
Definition:
(S_n) does not converge to S in R, there exists e > 0, ...(change from previous def. for all to there exists or vice versa)... |S_n - S| >= e.
Definition:
S_n does not converge iff , for all S in R, the previous def. holds.
Lemma: (last paragraph of sec. 7) Limits are unique.
If S_n converges to S (S_n -> S) in R and S_n -> t in R, then t = S.
Proof: pg. 37
Prof. Aside: for all e > 0, there exists N_s in NaturalN s.t. n > N_s => |S_n - S| < e/2, there exists N_t in NaturalN s.t. n > N_t => |S_n - S| < e/2. using, triangle inequality: |a+b| <= |a| + |b|, |s-t| = |S-Sn+Sn-T| <= |S-Sn| + |Sn-t| = |Sn-S| + |Sn-t| < e/2 + e/2 = e =>
0<=|S-t| < e => |S-t| = 0.
(Given e > 0, let N = max{Nt, Ns} given above, then n>N...)
-----------------------------------------------------
Section 8: Formal proofs
Ex. 1. lim 1/n = 0 = S by definition
Pf. given e>0, let N = 1/e, then n > N => 1/n < 1/N = e, so |Sn - S| = |1/n - 0| = 1/n < e. QED.
09/09/2015
Section 9
Limit Theorems
Theorems *9.1, 9.2, 9.3, *9.4, 9.5, 9.6, *9.7
09/10/2015
Theorems: *9.8, *9.10,
Exercise 9.9:
a). Use definition of S-> infty (S_n > M).
Section 10 - Cauchy - Monotone Seq.
===================================
Def. 10.1: Sn is increasing/non-decreasing iff S_n <= S_n+1 for all n. It is decreasing/non-increasing iff Sn >= S_n+1, for all n.
If it is either increasing or decreasing, it is called monotone:
(first edition of the book uses non-decreasing/non-increasing, thus, check definitions in different situations)
Theorems: *10.2 - 10.6
09/16/2015
HW 1 graded. Interesting note:
#7.4(b) Find rational sequence that has irrational limit.
Fibonacci sequence (facts by prof.: originated in population count of rabbits, in pairs unit; also relation to the golden ratio).
Definition 10.6, theorem 10.7, Def. 10.8 Cauchy sequence.
I should create Memrise version of index cards for these def. & theorems.
09/21/2015
lim Sn = sup_N inf_(n>N) {Sn}
lim Sn = inf_N sup_(n>N) {Sn}
10.8 Cauchy Sequence definition:
Lemma 10.9 Convergent => Cauchy
Proven.
Lemma 10.10 Cauchy => bounded
Proven.
*Theorem 10.11 Cauchy IFF convergent.
Proven.
Section 11 Subsequences
Def. 11.1
Theorem 11.2
*Theorem 11.3, 11.4
09/28/2015
, then either 
... I'll stop copying here...search proofs online.
Theorem 11.5 (BWT= Bolzano-Weierstrass Theorem):
Every bounded seq. has a conv. subsequence.
Alternate prove given in class:
(Sn) bdd. implies
Def. 11.6 shown.
Theorem 11.7 shown.
Theorem 11.8 shown: The set of subseq. limits of a given seq. (Sn) is:
a) not empty set. by 11.7
b) sup S = lim sup Sn, inf S = lim inf Sn (obvious)
c) lim Sn exists iff card(S) = 1.
ex. Sn = cos(nπ/2).
Theorem 11.9 shown.
09/30/2015
We few students chatted before the class. I learned that Luke Rawlings (adjunct) would be a good professor to take.
Quiz 1 on 10/14/2015
Sections 7-12
Prof: lim inf & lim sup apply to sequences; inf & sup apply to sets.
Theorem 12.1 (Sn) -> s positive real # and (tn) is any seq. then lim sup s_n*t_n = s* lim sup t_n. (compare theorems 9.4 & 9.9)
Proven in class.
Must read. Prof. said there's assumption in text that Sn is positive, something like that.
Theorem 12.2
10/05/2015
Section 13 (sec. 21, 22 are succession to 13)
Exercise 13.1 mentioned.
Definition 13.2 shown.
Cauchy seq <=> convergent is no longer true for general metric spaces:
Convergence implies Cauchy. But Cauchy does not implies convergence in general metric spaces, unless it is called a complete metric space.
Note: We always start with metric space in topology.
Discrete metric mentioned.
Theorem 13.5 BWT shown.
10/07/2015
Def. 13.6: Let (M,d) be a metric space + O \subset M, then O is open if \forall x \in O, \exists r > O, s.t. y \in O whenever d(x,y) < r. x \in S, r > 0
{y \in S: d(x,y) } < r} B_r(x) = neighborhood of radius r centered on x. i.e. x is an interior pt. of O, ie. \exists r > 0 s.t. d(x,y) < r. Lemma: O is open <=> the interior of O = the set of interior points = O.
F \subset M is closed if F^c = M \setminus F is open.
Discussion 13.7 mentioned (as "theorem"), note in iv) finitely, not infinitely. Because for (-1,1), {0} is closed, if intersect infinite collection of open sets.
Thus, when M is closed set, (consider at 13.7)
i). M is closed. (M^c = empty set <- open) ii).empty set is closed iii). intersect of any collection of closed sets is closed (de morgan's law applied) iv). Union of finite collections of closed sets is closed. Def. 13.8: The closure of a set E \congruent \bar{E} is the intersection of all closed sets containing E. Boundary... if a set is closed, it equals its closure. if it is opened, it equals its interior. Clopen = sets that are both open and close. Topology dictates this. HW do only 13.3, 5, 9, 10.
10/14/2015
Exercise 13.7. Extra credit to proof if it works for open ball in this case.
Notation for boundary:
Theorem 13.9
Example:
Theorem 13.10
On Cantor's Set
A set S is dense is in a set M iff closure S = M.
Example: closure (Rational set) = Real set
10/19/2015
Def. 14.3 Cauchy Criterion.
Theorem 14.4, 14.5 Corollary. to 14.9
15.1-3
Not in book:
is a bijection & 
, then, 
is called a rearrangement of 
.
Rearrangements of series
Definition: A series is conditionally convergent if it is convergent but not always convergent Ex.
if
Theorem: Any conditionally convergent series can be rearranged to converge to any real #, or to diverge (including to
).
Example:
= 1/1 - 1/1 + 1/2 - 1/2 + 1/3 - 1/3 + ... 
0.
= 1/1 + 1/2 - 1/1 + 1/3 + 1/4 - 1/2 + 1/5 + 1/6 - 1/3 + ... 
0.
Rearrange to
10/21/2015
Since last week, prof. used online notes: http://1drv.ms/1KlFihR
First 10 min of class Subbed by Giancarlo Paolillo
Section 17.
Continuity of real-valued functions of a real variable
17.1 Def.
17.2: (Tom Lehrer has a song about it: Epsilon delta)
17.3. See notes
17.4-5
Section 18
IVT (heuristic notion given. Also, toll booth uses mean value theorem).
10/26/2015
Theorem 18.1 to Corollary 18.3
Question Needs to be solved: Theorem 18.2
Why the author use "f(a) < y < f(b) or f(b) < y < f(a)" and not "f(a) ≤ y ≤ f(b) or f(b) ≤ y ≤ f(a)".
10/28/2015
Theorem 18.4, Inverse Functions (MATH 32400)
18.5-18.6
Section 19.1 Def.
Prof: Note at examples 2-3.
19.2
11/02/2015
Theorem 19.4
19.5 - don't confuse with 19.6
11/04/2015
is read: limit, as x tends to a along S, of f(x). This is similar to superscript usage between sets in set theory. 
is the set of all functions from Y to X.
Same substitute professor in the beginning.
20.1
Interesting notation:
20.3, and then skipped to 20.9.
And then back to 20.4-5.
The corollaries are mentioned, we're supposed to read them ourselves, I think.
Skipped to 20.10.
11/09/2015
Chapter 4: Seq's & Series of functions (Linear Algebra, Differential Equations MATH 39100)
Section 23 - No HW, simple review.
Theorem 23.1 shown
Sec. 24, point-wise convergence, uniformly convergence.
Def. 24.1
Def. 24.2
11/11/2015
Theorem 24.3
Remark 24.4 (Proof to be done in HW# 24.12)
Definition 25.3 and 25.4
11/16/2015 - 11/18/2015
Finished sec. 26-28...
28.4 Chain Rule
Faulty proof discussed by substitute.
11/30/2015
Theorem 29.1-9
Skipped section 30.
12/02/2015
Midterm Status:
high: 97.5 low: 51.5 median: 70 mean: 72.1
1 90s, 4 80s, 4 70s, 5 60s, 3 50s.
12/14/2015 Monday: Quiz #2 (sec. 28, 29,
31, 32)Chapter 6, Integration
Sec 32
Def. 32.1
Lemma 32.2, 32.3
Theorem 32.5 - 32.9
12/07/2015
c and b 
d then b-a 
d-c )
33.1 mentioned.
33.2, proven
33.3, mentioned. (Recall inf (S) = -sup (-S); if a
33.4 mentioned.
33.5 Proven. Consider proof at http://www.ux1.eiu.edu/~kparwani/integrals.pdf
33.6-9, proven.
Final:
All sections tested before + HW of 33
Theorem 34.3
Theorem 34.1 (Approximate theorem shown)
Integrability does not prove continuity.