## Basic properties of R

Archimedean Property

- dense of Q in R: always can find a rational number between two real numbers
- step functions

least upper/greatest lower bound property

- one concludes the another
- min and max

Trigonometric Inequality

- use in limit proof

irrational numbers

- proof of irrationality of square roots
- proof of existence of natural number-th root of positive real number

## Sequence and Series

Sequence

recursive definition

- techniques

Convergence

limit definition

- negation

- limit conservative theorem/squeeze theorem
- Operational rules of limit
- boundedness and convergence
- Monotonic Convergence Theorem
convergence of subsequence

- Nk >= K
- all subsequences are convergent iff convergent

- Nested interval theorem
Cauchy Sequnce

a sequence is cauchy iff it is convergent

- a->b: construction of subsequence/boundedness of cauchy sequence
- b->a:TI
- cauchy iff limSup = limInf

- cauchy sequence are bounded
applications

- approximation
real numbers are limits of cauchy sequences

- definition of real exponents

- the negation

LimInf/ LimSup

- Necessary Condition of Existence
Construction of an and bn

- monotonicity of an/bn
- convergence of an/bn

Construction of subsequence that converge to A/B

necessary condition: bounded

- B-W theorem: all bounded sequences have a convergent subsequence

technique

- |Ank-a|<1/k

conclusions

- weaker limit conservative theorem(very useful)

Series

convergence tests

- Cauchy's Criterion
- necessary: an goes to 0
comparison test

- absolute convergence indicates convergence
- greater than 0 and bounded by the bigger term\, by MCT\, convergent

root test(limSup)

- a<1\, a>1 are informative. a ->infinity is not

ratio tests(limSup)

- a<1 is informative only
- if liminf >1\, diverge

power series

- radius of convergence
e

series definition

- irrationality

limit definition

- limit conservative theorem

Convergence of summation by parts

- alternating series test

Convergence of catchy's Product

- one abs converge/ one converge

Convergence of rearrangement

converge to original value if abs converge

- gema(n) converge to +infinity

## Point-set Topology

set theory

- finite/infinite
countable/uncountable

- Z\, Q are countable
- union of countable set is countable
- bijection to N

definition of section and union

sequence of set

- union and intersection: definition

points

definitions

- neighborhood definition
- limit
- boundary
- exterior
- interior
- adherent

- relations

named sets(metric)

axioms of distant function

同一

- d(x\,y) = 0 <-> x= y

- 对称
- 三角

- open/close sets
- bounded/unbounded set
dense set in Metric space

- Q is dense in R by (AP)

compact sets

definition of open cover

- for mark a of each subset\, a can belong to a uncountable set

compact sets are closed

- by picking r(x) < d(x\,p)/2

compact sets are bounded

- use all neighbors of each element to construct an open cover

closed subset of a compact set is compact

- (A^c U open cover of A ) is an open cover of X

- H-B theorem: for a subset of R^n\, it closed and bounded iff it is compact
- techniques involved: chose Nr(p) for p in A

connected/disconnected set

- polygon theorem
a real interval is connected

- connected subset of R is an interval

- non-empty open disjoints

## Continuity

epsilon-delta definition in metric space

- must converge to values in the metric space(range)

convergence criterion of general metric space

- construct sequence that converge
- from general space to R^k\, k>=2
- from R^k to R (partial convergence)

continuity

- definition: limit points involved
open/closed set and continuity

- reverse image of open/closed subset of continuous function

connected set and continuity

- intermediate value theorem

compact set and continuity

- extreme value theorem

uniform continuity

- continuity and uniform continuity
uniform continuity and boundedness

- uniformly conti in (a\,b) -> bounded

compactness and uniform continuity

- continuous & compact domain -> uniformly continuous

discontinuity

side-limit

side limit and monotonicity

existence of side limit

- monotonic: both exist

- monotonic: convert to Sup and Inf
inequality of x\, y

set of dis continuity is at most countable

- (AP)/D injective to S\, S injective to Q

two types of discontinuity

- first kind: two side-limits both exist
- second kind: not both exist

## Differentiation

derivative

- operational
geometric

- interpretation: error decreases faster than variable

differentiable and continuous

- tools: linear inequality

local extrema

- def: delta language
interior + differentiable: derivative=0

- tool: limit conservative theorem

MVT

- condition: 1)continuous on [a\,b] 2)differentiable on (a\,b)
- tools: 1) Extreme value theorem 2) Construction of single variable function
conclusions

f(a)-f(b)=(a-b)f'(x)

monotonicity iff derivative has fixed sign

- condition: open interval\, differentiable

- constant iff derivative =0
R to R^k type : inequality

- tools: 1) EVT 2) Construction 3) Cauchy's inequality

injection and derivative

- f'(x) = 0: f is injective

L'Hospital's Rule

- tools: MVT\, injection\, LCT
condition

- existence of lim(f'/g')= L (can belong to R or infinity)
- f\, g differentiable on (a\,b)
- g'(x) \!= 0

two cases

- lim f(x)=0\, lim g(x)=0
- lim g(x) = infinity

conclusion

- lim (f/g)(x) = L

Taylor's Formula

- tools:

discontinuity of derivative

- special property: another version of Intermediate value
- discontinuous => second type

## Rhimann Integral

conditions

- defined on closed interval

inequality

- U(P\, f) >= L(P\, f)
inf U >= sup L

- taking common refinement

- f> g - > integration aslo true
- upper/ lower bound inequality

operational rules

f+g over I

- sup(A+B)<= SupA + SupB/ infA+B < = infA + infB

(f over I1) + (f over I2) = f over (I1 U I2)

- taking refinement of single point and P

f in R([a\, b]) -> |f| in R([a\, b])

- Sup f - Inf f >= Sup |f| - Inf |f|

f in R([a\, b]) -> f^2 in R([a\, b])

- EVT\, |f|<=M\, Sup f - Inf f >= Sup |f| - Inf |f|

f\, g in R([a\, b]) -> f*g in R([a\, b])

- fg = 1/2(f+g)^2 + 1/2(f-g)^2

integrate by parts

condition

- f(u(x))
- u' and f are integrable
- u is strictly increasing

- conclusion: the validity of integration by parts
tool

- distance interpretation
- bridge between two denotations

MVT for integration

tools

IVT

- f(x) should be continuous

construction for g(x) involving upper/lower bound

- EVT

integrability and

- difference
sequence of partition

- f in R([a\, b]) -> exist (Pn) in Pa\,b s.t limit converge to Riemann integration

monotonicity

monotone -> Riemann integrable

- taking uniform partiton

continuity

defined on compact set (closed interval) -> uniform continuity

- taking Unif partition with length smaller than delta

differentiability

construction of integration funtion

- continuity complement
uniform continuity of F(x)

- by Lipschitz function\, taking k = sup f

condition: f conti at x

- connection between domain and f
- other wise F is not necessarily differentiable

tool: MVT for integration

- |c- x| < delta by theorem

FTC

anti derivative.of a continuous function is the integration

- attention: still holds if f have some 1st type discontinuities

## Sequence and Series of functions

- point-wise convergence
uniform convergence

- def and negation
unif convergent(necessary) =>

- every sequence
- norm of uniform convergence->0
continuous f at x

- if fn(x) is conti at x for all n

- interchange of limit sign
- cauchy for every x
calculus

integration

- condition: fn(x) riemann integrable for all n
conclusion

interchange of limit and integrate

- integrate term by term if sum function Sn(x) is unif convergent

tool

- construct the norm of uniform convergence to be epsilon(n)

differentiation

condition(fn(x) needn't be known to be unif convergent)

- at least one point-wise convergence
- fn(x) is differentiable for all n
- derivative of fn(x) is uniform convergent

conclusion

- fn - > f uniformly
- f'n - > f' uniformly
- differentiation term by term

tools

- MVT
- goal: to show quotient converges uniformly
step1 show fn converges unif

- f'n converge point wise

- step2 f'n converges unif

valid conditions for unif convergence(sufficient)

Weierstrass M test

- only for series

- every sequence
- cauchy for every x
- norm of uniform convergence

complete matrix space

complete if every cauchy sequence in it is convergent to a point in that set

- contraction theorem for unique fixed point

power series

- power series and uniform convergence
Weirstrass approximation

condition

- f is continuous on [a\,b](compact set)

conclusion

- there exists a sequence of polynomial functions that uniformly converges to f

Ascoli's Theorem

condition

- compact domain
- uniformly bounded
- uniformly equicontinuous(equicontinuous would be enough\, uniformity given by compactness)

conclusion

- there is a uniformly convergent subsequence of (fn(x))

steps

1. existence of a countable subset of Domain on which there is a uniformly convergent subsequence

- W-B thm: existence of convergent subsequence
- diagonal argument

2. point-wise convergence of that subsequence on the WHOLE Domain

- compactness: finite open subcover with radius < delta
- TI

3. uniformity of that subsequence on the whole Domain

- finite set -> existence of max(Si)

## differentiation of multi-variable functions

definition of Differential of a function(Jacobean matrix)

- dimension match: f: n -> m \, Df in n*m
- cauchy inequality for matrix product

differentiability

- original definition
a sufficient condition(for R^n to R functions)

- Df is continuous

continuously differentiable

- original : each entry is continuous
matrix norm

- epsilon-delta

Chain rule

condition

- both functions are differentiable at certain point

conclusion

- differentiable at that point
- D(g(f(a))) = Df(a)*Dg(f(a))

MVT(general case)

condition

convex domain

- euclidian space
- balls in R^n are convex

- bounded differential

conclusion

- f is lipschiz in D

- use definition of convexity\, can change the domain to [0\,1]\, then apply MVT(inequality) for R -> R^n

inverse function theorem

condition

- Df(x) invertible at a
- f is C1

conclusion

- there is an open ball U centered at a s.t. f is bijective on U and f^-1(x):= g(x) is C1 on f(U)
- can apply chain rule to get Dg(f(a))

steps

- 1. construct a radius and a certain function (involving fixed point) s.t. the function is a contraction on the ball of the radius(bijectivity of U )
- 2. to show f(U) is open. still need the fixed point function
- 3. differentiability of f -> differentiability of g
4. continuity of Df + bijectivity: continuity of Dg

- shaky

implicit function theorem

condition

- f(a\,b) =0
- Dxf(a\,b) invertible

conclusion

- 1. there exist an open set W\, a function g s.t. all y in W\, there is a unique x s.t g(y) =x and f(x\,y)=0
- 2. g is C1
- Dg(b) = -Dyf(a\,b)(Dxf(a\,b))^-1

- key: complete a new function to make it square\!
- if f(a\,b)=0 and Dxf(a\,b) invertible\, then x can be solved uniquely around y=b with small enough r/ y can be solved around x=a if Dy(a\,b) invertible

differentiation across integral sign

condition

- the partial derivative is uniformly continuous on an interval(including s)
- D2f(x\,s) exist for all x

conclusion

a new uniform continuity

- continuous-> discrete -> continuous

- interchanging of integral and differentiating sign

higher order derivative

condition

- all second partial derivatives are continuous\, i\,e. f is C2

conclusion

- D12f = D21 f

key: double MVT

- two differences + order of limit

quadratic approximation for Rn to R function

- condition: C2 function
- conclusion: existence of remaining quadratic form
application: second derivative test

- supportive definition: definite positive/negative/ indefinite

lagrange multiplier method

- condition: two C1 functions(constraint\, target)\, existence of constrained extrema
- conclusion: parallel relationship of derivative at a
- tool: use implicit function theorem with constraint function

## integration of functions of several variables

jordan measure

- def
zero measure theorem

- equivalent condition for zero measure domain
- continuous function over zero measure boundary domain is integrable

differentiable surface

- image of C1 function from small space's compact subset to large space
- differentiable surface has zero measure boundary

fubini's theorem

condition

- rectangle
- riemann integrable
- partially integrable

conclusion

- double integral equals riemann integration
- change of integrating order
- double integral in general domain

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