# The World of Real Numbers This will begin with a description of real numbers, completeness, compactness, and some topology—ultimately leading to integration and differentiation. ## Real Numbers These are the numbers you and I know: 0, 1, -1, 1.5, and even $$\pi$$.\ **Rational** numbers are ones we can represent as an integer divided by another.\ It turns out many numbers are **irrational** including $$\pi$$, $$\sqrt{2}$$, $$\sqrt{3}$$, and $$\sqrt{n}$$ for\ any $$n$$ that's not a perfect square ($$\neq$$ a number squared). > **Idea**: use divisibility to show no rational number squared is 2. > > 1. if $$\sqrt{2} = a / b$$, then $$2 = a^2 / b^2$$ $$\implies a^2$$ is divisible by 2 > 2. So $$a$$ must be divisible by 2, implying $$b^2$$ is also divisible by 2, a contradiction! More abstractly, the real numbers are uniquely determined by a handful of natural **requirements** on a set: * **field** (operations: +, x with identity, inverses, commutiativity, associativity, and distribution) * **order**: trichotomy (either >, < or =) * **completeness** (= least upper bound property) * every set with an upper bound has a *least* upper bound. * e.g., (0, 1) has 1 as an upper bound, but 1 is not in the set (0, 1) A nice consequence of this is the **triangle inequality:** $$| x + y | \leq |x| + |y|$$ for any $$x, y \in \mathbb{R}$$. > **Idea**: look at cases where x, y are positive (Spivak has a nice proof of this) It turns out even in $$\mathbb{R}^n$$, a nice property about order called **Cauchy Schwarz** holds: $$ |\vec{u} \cdot \vec{v} | \leq |\vec{u}| | \vec{v} | $$ why? The idea is to use $$\vec{u} \cdot \vec{v} = |u||v| \cos($$ angel between them$$)$$. > #### primary::Vector > > a vector is a point in $$\mathbb{R}^n$$ > > * it looks like $$(x\_1, x\_2, x\_3)$$ where each $$x$$ is in $$\mathbb{R}$$ > > dot product is an operation on two vectors defined as: > > $$(x\_1, x\_2, x\_3) \cdot (y\_1, y\_2, y\_3) = x\_1y\_1 + x\_2y\_2 + x\_3y\_3$$ Finally, how many real numbers are there?\ Cantor showed there are actually uncountably many---you can never come up with a way to write a list of all real numbers even if you had all the time in the world. > **Idea**: suppose someone claimed to have a list of all reals (this list would have to be infinite). Then, we write down a number that's different from each one on the list in the nth decimal place Nevertheless, the set of rationals is countable. Make a table: ![](https://divisbyzero.files.wordpress.com/2013/04/screen-shot-2013-04-16-at-9-23-14-pm.png) This is a great [TedEd video](http://ed.ted.com/lessons/how-big-is-infinity) discussing infinity and the reals. > #### warning::Infinity + or - > > is **not** in the set of real numbers **Fundamental Theorem of Arithemetic**: every integer can be factor into primes ### Set Theory A set of ordered pairs if called a **relation**. A **one-to-one** (or injective) function sends inputs to distinct outputs > f(a) = f(b) implies a = b A **onto** (or surjective) function hits all outputs with some input. **DeMorgan's Law**: $$(A \cap B)^c = A^c \cup B^c$$ > idea: draw a picture (venn diagram) to see it's true. ## Topology ### Distance We can define a more general way to measure distance between two points in $$\mathbb{R}^n$$ (or any vector space), called a **norm**, denoted || \* ||. A norm is (for $$x, y \in \mathbb{R}^n$$): 1. *positive* (||x|| > 0, for all x in the set); *definite* || x || = 0 if and only if x =0 2. ||ax|| = a||x||, for any a in $$\mathbb{C}$$ 3. *triangle*: $$| x + y | \leq | x | + | y |$$ Common examples are Euclidean distance (L2 norm), Taxi Cab (L1 norm), and sup-norm (returns the peak value across all dimesions of vector; sup is the least upper bound, which isn't always in the set of interest). > remember **vectors spaces** are sets with nice properties about addition and scalar multiplication Norms in this context will later generalize to classes of functions with nice integration properties in measure theory. > #### primary::L2 and Lp norms > > $$(\sum\_i |x\_i|^p )^{1/p}$$ > > In the case of L2, p = 2 (this will show up again in measure theory). More generally we can talk about metric spaces, rather than just $$\mathbb{R}^n$$ or a vector space with a norm. A **metric space** is any set M with a distinace metric, d: MxM -> \[0, ∞) such that 1. symmetric: \_d(a, b) = d(b, a) for a, b in M 2. *definite*: d(a, b) = 0 if and only if a = b 3. triangle: \_d(a, c) ≤ d(a, b) + d(b, c) Many of the topological properites we'll explore applies to general metric spaces: all you need is a set and a reasonable way to measure distance. > #### info::topological space > > is a set M and a collection of subsets, S, that contains the empty set & M, is closed under union, and finite intersections. > > e.g., the open sets of a metric space always form a topological space! ### Neighborhoods With a notion of distance (often, we use L2 or Euclidean distance), we can define **neighborhoods** or **open balls** around a point: **B(x; r)** is a ball of radius r around x = set of points which are within a distance r of x (strictly < here) > formally, $$B(a; r) = { x \in set : || x - a || < r }$$ A set is **open**, if for any point, there is an open ball (of any radius) *entirely contained* in the set. > It turns out even open set is the union of open balls (potentially infinitely many). Why? > > 1. for each x in the Open set, there is a radius r such that B(x, r) ≤ Open set > 2. the union of all these B(x, r) (for each x in the open set) covers the open set > 3. but each B(x, r) is contained in the open set ==> U B(x,r) = Open set! A set is **closed**, if the set's complement is open. > #### warning::closed and open > > are not mutually exclusive; a set can be both! (for example $$\mathbb{R}$$ and $$\emptyset$$ ) An **accumulation** (or **limit point**) of a set is one where every open ball centered at the limit point contains at least one other point in the set. Alternatively, we can think of a limit point as the limit of some sequence of points in the set. This leads to another characterization of **closed** sets: sets that contain all their limit points. #### Unions and Intersections of Open and Closed Sets | U Open is Open | Intersection of Closed is Closed | | -------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------- | | For any x in U open, x is contained in some open set. So there is B(x, r) entirely contained in some open set, hence contained in the U open sets. | similar lidea to open sets | > in pictures, I think of the union of a bunch of open intervals, which is going to be open. For closed, I think about the intersection of a bunch of closed intervals \[0, 1], \[1\\/4, 3\\/4], ... this will be closed, because worst case, it'll contain a single point. While, not rigorous, this helps me imagine what's going on. ### Compactness A metric space is **compact** if every open cover has a finite subcover. An **open cover** is a union of open sets containing the metric space. It turns out there's a relationship between compactness and sequences in a space. If every sequence in space has a convergent subsequence, we say the the space is **sequentially compact**. It turns out **sequentially compact** and **compact** are equivalent in any metric space! > #### warning:: In Topological Spaces > > sequentially compact and compact are **not** equivalent A **compact** metric space is always **complete**, meaning every Cauchy sequence converges. > **idea**: Given any Cauchy sequence, a\_n, we can always find a convergent subequence (by sequential compactness). Since the sequence is Cauchy, the terms must be arbitrarily close to the convergent subsequence, hence converge. ### Cantor's Intersection Theorem In a **compact space** X, if closed, nonempty sets C1, C2, ... are nested: C1 $$\supseteq$$ C2 $$\supseteq$$ C3 ... then $$ \cap C\_n \neq \emptyset $$ > **Idea:** suppose $$\cap C\_n = \emptyset$$, then\ > 1\. Let $$O\_n = X \setminus C\_n$$ for each n\ > 2\. Then U On is an open cover for X => there is a finite subcover\ > 1\. say $$\cap\_{n=1}^k O\_n$$\ > 3\. Yet, because C\_n are nested, only a single O\_i covers X\ > 4\. Then $$C\_i = X \setminus O\_i = \emptyset$$, a contradiction! This will be used to show two famous theorems: Bolzano-Weierstrass and Heine-Borel. > #### warning:: the set of reals > > is **not** compact ### Bolzano-Weierstrass Theorem Every bounded sequence in $$\mathbb{R}^n$$ has a convergent subsequence. > **bounded** means contained in a ball of finite radius Why is this true? We'll chop our bounds in half and use the nested intervals theorem: ![](/files/-LM8yMz1l6OpwsOlm3BM) ### Heine-Borel A set K in $$\mathbb{R}$$ is **compact <====>** it's **closed** and **bounded** > #### primary::Tool > > **a closed subset, C, of a compact, K, set is compact** > > 1. Let U O be an open cover of C > 2. U O U {K C} is open and covers K > 3. there is a finite subcover of K, hence one for C With this tool we can now show why Heine-Borel is true > idea: > > 1. **=> Bounded**: suppose not > 1. there exists some unbounded sequence, an > 2. every subsequence is also unbounded, contradicting Bolzano-Weierstrass > 2. **=> closed**: idea show K already contains its boundary > 1. choose x in K's closure > 2. there is a sequence xn in K converging to x > 3. xn is Cauchy, implying xn must converge to a point in K > 1. hence x is in K > 3. **<= Compact**: 1. K is inside some interval \[-M, M], because it's bounded 2. Since K is closed and \[-M, M] is compact, K is closed 1. by our Tool above! \[notes transcribed] ## Big Picture What have we accomplished? We showed x, y, and z... **TODO**: * understand abstraction level between various spaces (metric, vector, function) and their corresponding distance measures. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://markibrahim.gitbook.io/why-calculus-works/real_analysis/reals_topology.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. 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