Lecture Notes and Videos on Introduction to Analysis I
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Copyright: I (Bala Krishnamoorthy) hold the copyright
for all lecture scribes/notes, documents, and other
materials including videos posted on these course web
pages. These materials might not be used for commercial
purposes without my consent.
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Scribes from
all lectures so far (as a single big file)
| Lec | Date | Topic(s) | Scribe | Video |
| 1 |
Aug 19 |
syllabus,
notation for logical statements, contrapositive proof,
proof by contradiction, proof by induction
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Scb1
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Vid1
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| 2 |
Aug 21 |
sets and operations, union, intersection, distributive laws, set
difference, De Morgan's laws, Cartesian product
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Scb2
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Vid2
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| 3 |
Aug 26 |
problem on Cartesian product, families of sets, set
operations over families, functions, composition,
pre/image
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Scb3
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Vid3
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| 4 |
Aug 28 |
pre/images commute with \(\cup,\cap\) or not (for
\(\cap\)), injective, surjective \(f\)'s, relation,
equivalence relation, partition
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Scb4
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Vid4
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| 5 |
Sep 2 |
\(\{[x]\}_{x \in X}\) under \(\sim\) partitions \(X\),
equivalence classes of fruits, countability, Cartesian
product of countable sets
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Scb5
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Vid5
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| 6 |
Sep 4 |
\(\mathbb{Q}\) is countable, \(\mathbb{R}\) is
uncountable, triangle inequality (two versions),
convergence, \(\{x_n\}\)\(\to\)\(a \Rightarrow\)\(
\{Mx_n\}\)\( \to\)\( Ma\)
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Scb6
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Vid6
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| 7 |
Sep 9 |
convergence in \(\mathbb{R}^m\), continuity, \(f_i\)
continuous \(\Rightarrow f_1 + f_2 - f_3\) is,
\(g(x)\) continuous & \(g(a) \neq 0 \Rightarrow
1/g(x)\) is
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Scb7
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Vid7
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| 8 |
Sep 11 |
completeness, bounded set, \(\mathbb{R}\) complete but
\(\mathbb{Q}\) is not, monotone/bounded sequence, sup,
inf, lim sup, lim inf
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Scb8
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Vid8
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| 9 |
Sep 16 |
\(\lim\sup\)\(/\)\(\inf
a_n\)\(=\)\(b\)\(\Leftrightarrow\)\(\lim
a_n\)\(=\)\(b\), Cauchy sequences, continuity with
sequences, intermediate value theorem (IVT)
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Scb9
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Vid9
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| 10 |
Sep 18 |
proof of the IVT, subsequence, Bolzano-Weierstrass
(BW) theorem in \(\mathbb{R}\), extreme value theorem
(EVT)
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Scb10
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Vid10
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| 11 |
Sep 23 |
Rolle's theorem, mean value theorem:\(\,\exists c \in
[a,b]\)\(\,:\,\)\(f'(c)\)\(=\)\((f(b)-f(a)/(b-a)\),
metric spaces, taxicab distance
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Scb11
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Vid11
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| 12 |
Sep 25 |
metric must be finite (\(d(x,y)\)\(<\)\(\infty\)),
metric spaces examples, distance between functions,
isometry, embedding
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Scb12
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Vid12
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| 13 |
Sep 30 |
convergence in a metric space, \(\epsilon\)-\(\delta\)
& open ball definitions of continuity of \(f : X \to
Y,\) discrete metric space
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Scb13
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Vid13
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| 14 |
Oct 2 |
interior, boundary, & exterior points; open and closed
sets, interior \(A^{\circ}\) and closure \(\bar{A}\)
of set \(A\), midterm review
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Scb14
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Vid14
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| 15 |
Oct 7 |
Midterm exam
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| 16 |
Oct 9 |
\(\overline{B}(\mathbf{a};r)\) closed,
\((\bar{A})^{\rm c}\)\(=\)\((A^c)^{\rm o}\),
continuity w/ open sets: \(\forall V\)\(\ni\)\(
f(x_0), \exists U\)\(\ni\)\(x_0 : f(U) \subseteq V\),
convergent\(\Rightarrow\)Cauchy
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Scb16
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Vid16
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| 17 |
Oct 14 |
complete metric spaces, \((A,d_A)\)
complete\(\Leftrightarrow\)\(A \subset X\) closed,
contraction, Banach's fixed point theorem (BFPT)
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Scb17
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Vid17
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| 18 |
Oct 16 |
BFPT problem, subsequence in \((X,d)\), compact subset
of \((X,d)\), compact\(\Rightarrow\)closed & bounded,
converse in \(\mathbb{R}^m\)
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Scb18
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Vid18
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| 19 |
Oct 21 |
compact \(\Rightarrow\) complete, compact sets under
continuous \(f, f^{-1}\), EVT, \(K\) cpct
\(\Rightarrow\)\(f^{-1}(K)\) closed, totally bounded
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Scb19
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Vid19
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| 20 |
Oct 23 |
totally bounded\(\Rightarrow\)bounded, open cover
property (OCP), OCP\(\Rightarrow\)compact,
\(f(x)\)\(=\)\(\sup\{r|B(x,r) \subset O\}\) continuous
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Scb20
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Vid20
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| 21 |
Oct 28 |
OCP\(\Leftrightarrow\)compact, proof, problems using
OCP, pointwise and uniform continuity (same \(\delta\)
for all \(x\)) of functions
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Scb21
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Vid21
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| 22 |
Oct 30 |
equicontinuous, \(\{f_n\}\) pointwise/uniform
convergence, continuous \(\{f_n(x)\}\)\(\to\)\(f(x)\)
uniformly\(\Rightarrow\)\(f(x)\) continuous
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Scb22
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Vid22
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Last modified: Thu Oct 30 16:52:07 PDT 2025
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