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Real Number 实数

Integer 整数
Rational Number 商数（有理数）：两个整数的商，分母非零
Real Number 实数：有理数的缝隙，使用Dedekind Cut，能在有理数的缝隙上定义出唯一的实数（Dedekind Completeness Theorem）。

实数完备的意义：实数集没有缝隙

实数完备性的六大等价定理

确界原理（supremum principle）：实数集的任意非空有上界的子集必有上确界（最小上界）
单调有界定理：单调有界数列必收敛
区间套定理：闭区间套必收敛到唯一实数
有限覆盖定理：闭区间的开覆盖必有有限子覆盖
聚点定理（Bolzano-Weierstrass Theorem）
柯西收敛准则：数列收敛的充要条件是柯西列
- Cauchy Sequence:

Dedekind Definition

实数的戴德金定义

Dedekind Cut 定义的实数集: 实数集被定义为所有的戴德金分割：

partition of into two sets and such that:

Each element of A is less than B
A contains no largest element

Another Interpretation of this definition is: every real number is defined as a subset of rational number set .

Dedekind Completeness Theorem

Proposition：实数集的有序分划（、）确定一个唯一的实数

在实数集中，非空有上界的集合，必定存在一最小上界（Supremum）

集合中所有实数对应的戴德金分割左集的并集：

是一个戴德金分划（是一个实数）

是一个有理数集：都是有理数集
非空非：有上界，因此，因此，
向下封闭性/有序性：任意，如果，那么，又，，所以，或
无最大元：

是的一个上界：

是的最小上界：

对于的任意上界，

实数即集合

在Dedekind Cut的定义里，实数是由分割定义的，因此这个分割的左集——比它小的有理数集合就定义了一个实数。

实数次序的定义

上确界原理证明思路

Dedekind Cut定义的实数是有理数集的有序分割左集，任意有上界的实数集都能转换为对应有理数分割左集的并集。实数集等价的有理数分割左集并集也是几个有理数集，也对应一个实数，而其为实数集的最小超集，因此当有上界的时候，上确界存在且总是等于实数集等价的有理数分割左集并集。

Dedekind Completeness Theorem 证明思路

证明上确界原理：所有有上界实数集都有上确界，从而所有实数集分划左集都有上确界，从而有序分划能唯一确定一实数。
[!note] 为什么确界原理就是实数完备性
确界原理说任意有界实数集合都有上确界，就是说实数的任意分划能靠左集的有界性能确定唯一的一个实数。这样实数就不会像有理数一样，分划不能用来确定唯一的有理数，即“有洞”了。

有理数的稠密性：在任意两个实数之间，能找到有理数。

对于，，所以总有

Monotone Convergence Theorem

单调有界的数列收敛，存在。

根据上确界原理，设的上确界为

因此单调有界的数列必收敛，且收敛于上确界

Nested Interval Theorem

闭区间Compactness

有且仅有唯一的点，属于所有区间

闭区间列，，

左端点序列单调递增且有界，因此极限存在且
右端点序列同理，存在，且，由极限运算法则和区间长度趋向零得：

同时小于等于所有，同时大于等于所有，因此
唯一性：如果有两个都在内，那么左右端点序列就有不同的极限，与区间长度趋向零矛盾。因此唯一。

开区间套不能确定唯一的点：内的任意总存在足够大的使得从而

Finite Covering Theorem (Heine-Borel Theorem)

闭区间 Compactness

Open Cover：集合的开覆盖，，是开区间
Finite Subcover

闭区间存在开覆盖，则有有限覆盖：
假设不能被有限覆盖，二分，选择一不能被有限覆盖的区间作为，继续此操作构建闭区间套，从而由闭区间套定理，确定唯一的实数，而由于覆盖，，由于开覆盖，，从而被定义成不能被有限覆盖的区间被有限覆盖了，矛盾。

Bolzano-Weierstrass Theorem

聚点定理

有界数列必有收敛子列

有界无穷数列必有聚点（收敛子列）

证明
令，为二分的中包含无穷项的一半，以此构造闭区间套，从而能确定唯一的点。从每个中选择，则由夹逼定理：

Cauchy Convergence Criterion

柯西收敛准则

实数数列收敛的充要条件是该数列为柯西数列(Cauchy Sequence):

proof:

必要性

显然，数列若收敛，必为柯西列

充分性：

柯西列有界，因此存在收敛的子列收敛于 Bolzano-Weierstrass Theorem，选取子列时下标随着增大而增大且

因此

对于任意，找到使得柯西列差值的距离、收敛子列与a距离都小于的，选择，则

Real Number

If by selecting a small punctuated neighborhood of , a variable determined by can close enough to a fix value in any level, then that variable is converge to that fix value.

Definition of Limits

Limits of Sequence

is saying:

Limits of function

the limit of of as approaches

it means that the value of the function can be made arbitrary close to by choosing x close to

is saying:

Heine's Theorem

人话：所有路径都收敛于一个值，那么就在该区域无死角收敛了。

证明：

如果函数收敛，自然每一条路径都收敛
如果函数不收敛，那么能够构造不收敛的坏数列

Sign-Preserving Property

If a function has a limit on , then there's a punctured neighborhood of , one that set, keeps the same sign with the limit

Suppose

Set , then , ,

So as when

Squeeze Theorem

if , and and has same limitation on , then also has limitation on

Order Property of Limits

If , and , then

if , then for , :

which contradicts with . So

Local Boundedness

Local Boundness

If , is bounded in some punctured neighborhood of

Rules with Real Number Operations

Constant Rule

swappable to scalar product

Proof

with , , thus:

Sum Rule

distributive to function addition

Proof: (triangular Inequality)

with ,
with ,
Thus

Product Rule

distributive to function product

Proof:

involve a trick of constructing and
be aware that is locally bounded due to the Local Boundness

with , , thus
with ,

so, with ,

Quotient Rule

distributive to function devision

Proof

key steps:

step1: Proof

step2: use distributive to function product:

Detail Proof

This proof is divided into two steps:

proof and reverse triangular inequality
product rule

with , , which saids:

with ,

Thus:

applying the product rule:

Composition Rule

composition rule

if is continuous, then

Limits

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Definition

if a function is continuous on , it's saying that for any , s.t.:

Boundedness Theorem

If a function is continuous on a close-interval

, then its bounded on that interval.

Proof with Bolzano-Weierstrass Theorem.

Suppose that: the function boundless on

If the function is boundless, loop this process, starts with :

bisect the interval with
select the boundless half or
select a point making this half boundless to build a sequence
continue this process with

And finally we have a infinite bounded sequence and an unbounded sequence s.t.:

converge to some number (Bolzano Weierstrass Theorem)
is unbounded, which contradicts to (Heine Theorem)

So the function is bounded on

Intermediate Value Theorem

IVT

If a function is continuous on a close-interval , then for any intermediate value , there's some value making that

Proof

Build nested intervals with this process:

bisect the interval with
if , stop.
select the half making
continue this process

So we get a nested interval , which locate exactly one real number , and (Nested Interval Theorem). As the function is continuous on , with Heine's Theorem:

So we have another set of nested intervals that locate a unique number, which can only be since . (Nested Interval Theorem)

Extreme Value Theorem

If a function is continuous on , then there's a maxima and minima in .

As Boundedness Theorem said, the function is bounded, so the function has a supremum and infimum, w.r.t. Dedekind Completeness Theorem. If the supremum is .

Suppose . is also a continuous function, thus it's also bounded. Suppose is an upper bound of on :

so is also an upper bound for on , which contradicts to the setting that is the supremum. So there must be some value s.t. .

Mean Value Theorems

Fermat's Lemma

derivative on maxima / minima must be zero

Proof:

Suppose is a maxima.

It said that:

Rolle's MVT

if function is continuous on and , then:

A function is continuous on , then with Extreme Value Theorem, there must be a maxima , and then with Fermat's Lemma,

Lagrange MVT

If function is continuous on , then:

, which saids the derivative on some intermediate point concludes the change of endpoint.

Proof:

build a flat function containing to apply Rolle's MVT:

So that

So with Rolle's MVT,

Cauchy MVT

if function are both continuous on , then:

construct a function whose derivative is in the form that fit the theorem:

so by Rolle's MVT:

MVT for Integrals

If function is continuous on , then the mean integral is an intermediate value of

is continuous on , so with Boundedness Theorem, is bounded:

then with IVT:

Fundamental Theorem of Calculus

if is continuous at

accumulation function is one of the antiderivative

Integration is difference:

Proof:

Prove it by MVT for Integrals:

So any original function of , denoted by , can be represented by adding some constant to :

substituting with , we know the constant is

substituting with , then the integration of from to is the difference of the value of the endpoints of any antiderivative :

Prerequisite of differentiable

it must be continuous to be derivable, or equivalently, differentiable.

if is differentiable, then is also an infinitesimal -- which implies that is continuous.

Continuous Functions

Definition of Series

Series is an infinite sum, or the limit of Partial Sum

sum of numbers

sum of functions

Different Types of Series

Series of number

Geometric
Harmonic
P-Series
Alternating

Geometric Series

Harmonic Series

P-Series

Alternating Series

Power Series

Power series: series of power functions: sum of powers of

convergence/divergence

Tests:
Ratio test
Integral test
Compare test

To proof the image of the function is the limit of the Taylor series, write the series in the form of the sum of partial sum and Remainder:

Divergence Test

Ratio and Root Tests

Comparison Tests

Integral Test

Alternating Index

odd and even numbers:

product is odd only when all factors are odd

sum is odd only if there’s odd number of odd terms

Operations of Series

Sum Rule

Proof

Given:

So:

Limit rule

distribute calculating limitation w.r.t addition

Series

为什么用单位圆和直角坐标系定义三角函数

用直角三角形定义的话角度被限制在0至，用角边和单位圆交点坐标定义则能轻松将三角函数的定义域扩展到任意实数值。

关于

我们现在不把它当作圆周率，而是平角的角度

单位圆定义三角函数

由平面直角坐标系
单位圆与x轴正半轴交于点，以直线为一边，另一边与单位圆交于，且的坐标为，则角的正弦和余弦值为。换句话说，正弦函数和余弦函数将角的大小映射到的坐标分量上。的坐标也可记为

诱导公式

Note

将所有常数写在前面方便后续套用改变运算顺序或消掉常数

大小为的角与单位圆的交点与关于x轴对称，因此有：

大小为的角与单位圆的交点与关于原点对称，由此有：

大小为的角与单位圆的交点与关于对称，由此有：

余弦定理

仅使用勾股定理，分锐角三角形、钝角三角形、直角三角形三种情况（钝角三角形证明部分需要利用诱导公式）可证明对于任意由点ABC围成的平面三角形，记角ABC的对边长度为，角A大小为，有：

和差公式

余弦差公式

作图，作出大小为, , 的角与单位圆的交点，利用两个角度大小为的弓形的弦长相等得

Note

这样的标号应看作一个复合函数在的值

其余和差公式

其余和差公式的推导不过是运用实数加减法运算性质、余弦差公式、和诱导公式

由于正弦余弦函数和差公式的齐次性，可得的和差公式

倍角公式

令，代入上面的和差公式容易得到：

半角公式

将倍角公式反过来就是半角公式：

积化和差

三角函数值的积可以化为三角函数值的和

可以发现和差公式的右侧的两项是很容易消去的其中一项的，因此能够有积化和差公式：

和差化积

既然积能等于和差，那么和差也能等于积。换元：

辅助角公式

辅助角公式是用反正切函数和正弦和角公式，将任意角度的正弦和余弦值的任意线性组合化为一个单一的正弦函数：

Series defined trigonometric functions

defined the and as a form of Tayler Series, with the heuristic of their geometric definition:

Trigonometry Functions

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Big Three Ratio

Big

: The Ceiling

means growth rate is upper-bounded by

This saids: there exists a constant and a constant , as , , or

, is bounded by

Example 1

Prove that

as , , which saids , it's bounded by

Big Theta Notation

Big Theta Notation: Same Growth Rate

implies has exactly the same growth rate as

Big Omega Notation

saids that , as long as

Little

little

: strict ceiling

means becomes insignificant compared to

rules for little o:

multiplying an constant doesn't change its order:
multiplying an infinitesimal make it an even higher order infinitesimal:
- is or
adding a higher order of infinitesimal doesn't change its order: if , then

Notations

symbol	Asymptotic meaning	read it out	The algorithm is...
		grows strictly slower than	little o of g of x
		grows no faster than	o of g of x
		grows at least as fast as	omega of g of x
		grows strictly faster than	little omega of g of x
		grows as fast as	theta of g of x

Summary

little means strict, big means no ... than
is "o" means upper-bound, is "omega" means lower-bound, is "theta" is the same order as

Asymptotic Notations

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Equivalent Infinitesimals

and are equivalent infinitesimals, written as saids that

oscillating function

and should not cross zero "near" , otherwise, the quotient is not defined.

Transitivity of equivalent

As for Product Rule, so the equivalent relation has transitivity:

Proof:

Equivalent Infinitesimal Trick

Infinitesimals

Definition

Riemann Sum

Riemann Integral

and , as long as , then

Then is the limit of Riemann sum

Properties

This is due to the definition of Riemann integral and linear properties of limits. Limits > Rules with Real Number Operations

Riemann Integral

Integral Definition of Logarithmic

推广指对数函数到实数

定义

因为该变上限积分有以下和有理数指对数相似的性质：

积 分 定 义 换 元 反 函 数 求 导

再用换底公式来定义，并定义为的反函数，则由以上结论容易有

指数函数也符合幂乘的运算法则：

因此有：

Euler's Number

自然底数

定义底数使得，即

由导数定义：

以及重要极限的存在性，因此：

称为自然对数的底数

Exponential and Logarithmic Functions

Tayler Series

Taylor Series: simulating utilizing its behavior on

Proof

Fundamental Theorem of Calculus

Integrate by parts Integration skills > integration by parts

Integrate by parts:

Suspect that:

Inductive Reasoning

Maclaurin Series

Maclaurin Series is simply Tayler Series center at :

Lagrange Remainder

By MVT for Integrals

So we can derived Lagrange Remainder from the Integral Remainder:

Example: computing

knowledge:

So we know well about around , and thus can compute around .

from .factorial import factorial

DERIVATIVES = [0, 1, 0, -1]

pi = 3.141592653589793

def sin(x: float) -> float:
    if x < 0:
        return -sin(-x)
    if x > 2 * pi:
        return sin(x % (2 * pi))

    res = 0
    for n in range(100):
        res += DERIVATIVES[n % 4] * (x ** n) / factorial(n)
    return res

Tayler Series

Dedekind Definition

Monotone Convergence Theorem