# Analysis

Analysis notes based mostly on Tao's *Analysis I* and
*An Introduction to Measure Theory*.

- Exercise: nested sequences of sets
- Perfect Sets
- Convergence of subsequences
- Heine-Borel Theorem
- Square root, expressed as the limit of a sequence
- The Cantor Set
- 18 analysis theorems
- 18 analysis theorems
- 18 analysis theorems
- 18 analysis theorems
- Monotone bounded sequences converge
- Archimedean property
- Cut property for real numbers
- Nested intervals property for reals
- The irrationality of \( \sqrt(2) \)
- Lebesgue measurability. Definition.
- Lebesgue measurable sets.
- Outer regularity. Theorem.
- Lebesgue outer measure (and Jordan inner measure) for any open set
- Outer Lebesgue measure of countable union of almost disjoint boxes
- Carathéodory type property
- Closures, interiors and Jordan measure
- Closures, interiors and Jordan measure
- Closures, interiors and Jordan measure
- Closures, interiors and Jordan measure
- Lebesgue outer measure is between the Jordan inner and outer measures
- Lebesgue outer measure. Finite additivity (for separated sets)
- Lebesgue Measure. Definition
- The 3 basic properties of Lebesgue outer measure
- Derivative of monotone functions
- Mean value theorem
- Rolle's theorem
- Differentiability on a domain
- Differentiability ⇒ continuity
- Differentiability at a point, definition
- Newton's approximation
- Limits at infinity (for continuous function)
- Monotonic functions
- The intermediate value theorem
- Uniform continuity
- Uniform continuity (sequence dual)
- The maximum principle
- Left and right limits
- Left and right limits and discontinuities
- Arithmetic preserves continuity
- Equivalent formulations of function continuity
- Equivalent formulations of function continuity
- Continous functions
- Calculating function convergence
- Function convergence at a point, definition
- Function convergence's equivalence to sequence convergence
- Limit laws for functions, proposition
- Arithmetic operations on functions
- Bounded sets (of reals)
- Closed set (of reals), definition
- Closure, definition
- Heine-Borel theorem for the line (Tao)
- Heine-Borel theorem for the line (Tao)
- Intervals (of the reals), definition
- Limit points and isolated points (of sets of reals), definition
- Maximal and minimal elements, definition
- Well-ordered sets
- Axiom of Choice
- The Continuum Hypothesis
- Cantor's theorem
- Power set axiom
- Bernhard Riemann's rearrangement theorem
- Infinite series and summation on infinite sets (summation laws III and IV)
- Order relations, visualized on graphs
- Series laws Ⅰ: finite series laws
- Fubini's theorem for infinite series
- Series on uncountable sets (a special case)
- Count all the things (countability propositions)
- Countable sets
- Series on countable sets, definition
- Cardinality of sets
- Ratio test
- The Root Test
- Rearrangement of infinite series
- Cauchy criterion, harmonic series and the Riemann-zeta function
- Geometric series
- Telescoping series
- The comparison test (and bounded series of non-negative numbers)
- Alternating series test
- Absolute convergence, and the absolute convergence test
- Zero tail, and the zero test, propositions
- Infinite series, definition
- Fubini's theorem for finite series
- Series laws Ⅱ: sums over finite sets
- Summation over finite sets, definition
- Bolzano-Weierstrass theorem
- Finite series, definition
- Subsequences and limits, proposition
- Subsequences, definition
- Completeness of the reals, theorem
- Squeeze test
- Zero test for sequences
- Supremum of sets of extended reals
- Limit superior and limit inferior
- Limit points
- Suprema and infima of sequences, definition
- Formal limits are genuine limits
- Limit laws
- Limit laws
- Limits of sequences
- Uniqueness of convergence, proposition
- Convergence of sequences of reals
- Cauchy sequences of reals
- \( n^{th} \) root of a real
- \( x^2 = 2 \). There exists a positive real whose square is 2. Proposition.
- Cauchy sequences
- Eventual ε-steadiness
- Existence of least upper bound, theorem
- Least upper bound
- Supremum
- Uniqueness of least upper bound, proposition
- Upper bound
- Reciprocals of real numbers
- Sequences bounded away from zero
- Equivalent sequences
- Eventually ε-close sequences
- Real numbers, the construction from Cauchy Sequences
- ε-close sequences
- Addition, the definition for natural numbers
- Bounded sequences
- Sequences
- ε-steadiness
- Tao's 5 axioms for natural numbers

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