A subset of a set BBB is a set AAA such that every element of AAA is an element of BBB. Formally, A⊆Bmeans∀x (x∈A⇒x∈B).A \subseteq B \quad\text{means}\quad \forall x\,(x\in A \Rightarrow x\in B).A⊆Bmeans∀x(x∈A⇒x∈B).Subset inclusion ⊆\subseteq⊆ is the basic relation for comparing sets. Many constructions in analysis (closures, neighborhoods, function spaces) are naturally expressed as subsets of larger ambient sets. ...

Substitution rule: Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous , and let φ:[α,β]→[a,b]\varphi:[\alpha,\beta]\to[a,b]φ:[α,β]→[a,b] be continuously differentiable and monotone on [α,β][\alpha,\beta][α,β]. Then ∫αβf(φ(t)) φ′(t) dt=∫φ(α)φ(β)f(u) du. \int_\alpha^\beta f(\varphi(t))\,\varphi'(t)\,dt=\int_{\varphi(\alpha)}^{\varphi(\beta)} f(u)\,du. ∫αβ​f(φ(t))φ′(t)dt=∫φ(α)φ(β)​f(u)du. (If φ\varphiφ is decreasing, the right-hand side automatically changes sign because φ(α)>φ(β)\varphi(\alpha)>\varphi(\beta)φ(α)>φ(β).) ...

Let (X,≤)(X,\le)(X,≤) be a partially ordered set and let S⊆XS\subseteq XS⊆X. An element s∗∈Xs^\ast\in Xs∗∈X is the supremum of SSS, written s∗=sup⁡Ss^\ast=\sup Ss∗=supS, if: s∗s^\asts∗ is an upper bound of SSS, i.e. ∀s∈S, s≤s∗\forall s\in S,\ s\le s^\ast∀s∈S, s≤s∗, and s∗s^\asts∗ is the least such upper bound: for every upper bound uuu of SSS, one has s∗≤us^\ast\le us∗≤u. Suprema are “best possible” upper bounds and are the key completeness feature of R\mathbb{R}R. In general ordered sets, sup⁡S\sup SsupS need not exist. ...

Supremum approximation lemma: Let E⊆RE\subseteq\mathbb{R}E⊆R be nonempty and bounded above , and let s=sup⁡Es=\sup Es=supE. Then: for every ε>0\varepsilon>0ε>0 there exists x∈Ex\in Ex∈E such that s−ε<x≤s, s-\varepsilon < x \le s, s−ε<x≤s, equivalently, there exists a sequence (xn)(x_n)(xn​) in EEE such that xn→sx_n\to sxn​→s. This lemma is used constantly to turn the abstract existence of sup⁡E\sup EsupE into a usable approximation statement (often in ε\varepsilonε–arguments). ...

A function f:X→Yf:X\to Yf:X→Y is surjective (or onto) if ∀y∈Y, ∃x∈X such that f(x)=y.\forall y\in Y,\ \exists x\in X\ \text{such that}\ f(x)=y.∀y∈Y, ∃x∈X such that f(x)=y. Equivalently, f(X)=Yf(X)=Yf(X)=Y (the image equals the codomain ). Surjectivity depends on the specified codomain YYY, not just on the rule x↦f(x)x\mapsto f(x)x↦f(x). Many constructions in analysis naturally produce surjections (e.g., quotient maps, parameterizations) and surjectivity is required for an inverse to be defined on all of YYY. ...

The symmetric difference of sets AAA and BBB is A△B:=(A∖B)∪(B∖A).A\triangle B := (A\setminus B)\cup(B\setminus A).A△B:=(A∖B)∪(B∖A). Equivalently, x∈A△Bx\in A\triangle Bx∈A△B iff (x∈A) ⊕ (x∈B)(x\in A)\ \oplus\ (x\in B)(x∈A) ⊕ (x∈B), where ⊕\oplus⊕ denotes exclusive-or. Symmetric difference measures how two sets differ and is useful as an operation on sets (it makes the power set of a fixed universe into an abelian group under △\triangle△). ...

A tagged partition of [a,b][a,b][a,b] is a pair (P,T)(P,T)(P,T) where: P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0​<⋯<xn​=b is a partition, and T=(t1,…,tn)T=(t_1,\dots,t_n)T=(t1​,…,tn​) is a choice of tags with ti∈[xi−1,xi]for each i=1,…,n.t_i\in[x_{i-1},x_i]\quad\text{for each }i=1,\dots,n.ti​∈[xi−1​,xi​]for each i=1,…,n. Tagged partitions specify where a function is sampled to form Riemann sums. Different tagging conventions (e.g., left endpoints, right endpoints, midpoints) yield different approximations for finite partitions. ...

Let fff be a real- (or complex-) valued function defined on an interval containing a∈Ra\in\mathbb{R}a∈R, and assume that f(j)(a)f^{(j)}(a)f(j)(a) exists for 0≤j≤k0\le j\le k0≤j≤k (see higher derivatives ). The Taylor polynomial of degree kkk of fff at aaa is Tkf(x;a):=∑j=0kf(j)(a)j!(x−a)j. T_k f(x;a) := \sum_{j=0}^k \frac{f^{(j)}(a)}{j!}(x-a)^j. Tk​f(x;a):=j=0∑k​j!f(j)(a)​(x−a)j.Taylor polynomials provide the canonical local polynomial approximation to fff near aaa. Taylor’s theorem quantifies the error via a remainder term . ...

Taylor’s Theorem (several variables, one standard form): Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rf:U\to\mathbb{R}f:U→R be of class Ck+1C^{k+1}Ck+1 on a neighborhood of a∈Ua\in Ua∈U. Using multi-index notation, there exists a remainder Rk(h)R_k(h)Rk​(h) such that for hhh sufficiently small (with a+h∈Ua+h\in Ua+h∈U), f(a+h)=∑∣α∣≤kDαf(a)α! hα+Rk(h), f(a+h)=\sum_{|\alpha|\le k}\frac{D^\alpha f(a)}{\alpha!}\,h^\alpha + R_k(h), f(a+h)=∑∣α∣≤k​α!Dαf(a)​hα+Rk​(h), and Rk(h)∥h∥k→0as h→0. \frac{R_k(h)}{\|h\|^k}\to 0 \quad \text{as } h\to 0. ∥h∥kRk​(h)​→0as h→0. Here α=(α1,…,αn)\alpha=(\alpha_1,\dots,\alpha_n)α=(α1​,…,αn​), ∣α∣=α1+⋯+αn|\alpha|=\alpha_1+\cdots+\alpha_n∣α∣=α1​+⋯+αn​, α!=α1!⋯αn!\alpha!=\alpha_1!\cdots \alpha_n!α!=α1​!⋯αn​!, and hα=h1α1⋯hnαnh^\alpha=h_1^{\alpha_1}\cdots h_n^{\alpha_n}hα=h1α1​​⋯hnαn​​. ...

Taylor’s Theorem (Lagrange remainder): Let fff be (n+1)(n+1)(n+1) times continuously differentiable on an interval containing aaa and xxx. Then there exists ξ\xiξ between aaa and xxx such that f(x)=∑k=0nf(k)(a)k!(x−a)k+f(n+1)(ξ)(n+1)!(x−a)n+1. f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}. f(x)=∑k=0n​k!f(k)(a)​(x−a)k+(n+1)!f(n+1)(ξ)​(x−a)n+1. ...