Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The boundary of AAA, denoted ∂A\partial A∂A, is ∂A:=A‾∖int⁡(A).\partial A := \overline{A}\setminus \operatorname{int}(A).∂A:=A∖int(A). Equivalently, ∂A=A‾∩X∖A‾\partial A = \overline{A}\cap \overline{X\setminus A}∂A=A∩X∖A​ (see closure and interior ). Equivalently again, x∈∂Ax\in\partial Ax∈∂A iff every open ball B(x,r)B(x,r)B(x,r) meets both AAA and X∖AX\setminus AX∖A. ...

Let (X,≤)(X,\le)(X,≤) be an ordered set and let S⊆XS\subseteq XS⊆X. The set SSS is bounded above if there exists an element u∈Xu\in Xu∈X such that ∀s∈S, s≤u.\forall s\in S,\ s\le u.∀s∈S, s≤u. Equivalently, SSS is bounded above iff SSS has an upper bound. Boundedness above is the hypothesis needed to speak meaningfully about sup⁡S\sup SsupS (and in R\mathbb{R}R, completeness asserts existence of sup⁡S\sup SsupS for every nonempty bounded-above set). ...

Let (X,≤)(X,\le)(X,≤) be an ordered set and let S⊆XS\subseteq XS⊆X. The set SSS is bounded below if there exists an element ℓ∈X\ell\in Xℓ∈X such that ∀s∈S, ℓ≤s.\forall s\in S,\ \ell\le s.∀s∈S, ℓ≤s. Equivalently, SSS is bounded below iff SSS has a lower bound. Boundedness below is the hypothesis needed to speak meaningfully about inf⁡S\inf SinfS. Examples: (0,1)⊆R(0,1)\subseteq\mathbb{R}(0,1)⊆R is bounded below (e.g. by ℓ=0\ell=0ℓ=0). {x∈R:x≥−7}\{x\in\mathbb{R}: x\ge -7\}{x∈R:x≥−7} is bounded below (e.g. by ℓ=−7\ell=-7ℓ=−7). The set {−n:n∈N}⊆R\{ -n : n\in\mathbb{N}\}\subseteq\mathbb{R}{−n:n∈N}⊆R is not bounded below.

Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R be differentiable on I∘I^\circI∘. Proposition: Suppose there exists M≥0M\ge 0M≥0 such that ∣f′(x)∣≤M|f'(x)|\le M∣f′(x)∣≤M for all x∈I∘x\in I^\circx∈I∘. Then for all x,y∈Ix,y\in Ix,y∈I, ∣f(x)−f(y)∣≤M∣x−y∣. |f(x)-f(y)|\le M|x-y|. ∣f(x)−f(y)∣≤M∣x−y∣. In particular, fff is Lipschitz on III and hence uniformly continuous on III. ...

Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn​) be a sequence in XXX. The sequence is bounded if its range {xn:n∈N}\{x_n:n\in\mathbb{N}\}{xn​:n∈N} is a bounded subset of XXX, i.e. if there exist x0∈Xx_0\in Xx0​∈X and M∈[0,∞)M\in[0,\infty)M∈[0,∞) such that ∀n∈N, d(xn,x0)≤M.\forall n\in\mathbb{N},\ d(x_n,x_0)\le M.∀n∈N, d(xn​,x0​)≤M.Boundedness is a minimal compactness-type hypothesis: in Rk\mathbb{R}^kRk, bounded sequences have convergent subsequences (Bolzano–Weierstrass ). ...

A subset SSS is called bounded in two common contexts: In an ordered set (X,≤)(X,\le)(X,≤), a subset S⊆XS\subseteq XS⊆X is bounded if it is bounded above and bounded below , i.e. if there exist ℓ,u∈X\ell,u\in Xℓ,u∈X such that ∀s∈S, ℓ≤s≤u.\forall s\in S,\ \ell\le s\le u.∀s∈S, ℓ≤s≤u. In a metric space (X,d)(X,d)(X,d), a subset S⊆XS\subseteq XS⊆X is bounded if there exist x0∈Xx_0\in Xx0​∈X and M∈[0,∞)M\in[0,\infty)M∈[0,∞) such that ...

C1C^1C1 implies differentiable: Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm. Suppose all first-order partial derivatives of fff exist on a neighborhood of a∈Ua\in Ua∈U and are continuous at aaa (equivalently, f∈HAHAHUGOSHORTCODE798s3HBHBf\in C^1 f∈HAHAHUGOSHORTCODE798s3HBHB near aaa). Then fff is differentiable at aaa. ...

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 $C^2$ . Corollary: For all a∈Ua\in Ua∈U and all indices i≠ji\neq ji=j, ∂2f∂xi∂xj(a)=∂2f∂xj∂xi(a). \frac{\partial^2 f}{\partial x_i\partial x_j}(a)=\frac{\partial^2 f}{\partial x_j\partial x_i}(a). ∂xi​∂xj​∂2f​(a)=∂xj​∂xi​∂2f​(a). ...

Cantor Intersection Theorem: Let (X,d)(X,d)(X,d) be a complete metric space and let (Fn)(F_n)(Fn​) be a sequence of nonempty closed sets such that Fn+1⊆Fnfor all n,F_{n+1}\subseteq F_n \quad \text{for all } n,Fn+1​⊆Fn​for all n, and diam⁡(Fn)→0as n→∞.\operatorname{diam}(F_n)\to 0 \quad \text{as } n\to\infty.diam(Fn​)→0as n→∞. Then ⋂n=1∞Fn\bigcap_{n=1}^\infty F_n⋂n=1∞​Fn​ consists of exactly one point. ...

The Cartesian product of sets AAA and BBB is A×B:={(a,b):a∈A, b∈B}.A\times B := \{(a,b) : a\in A,\ b\in B\}.A×B:={(a,b):a∈A, b∈B}.Cartesian products encode “simultaneous choices” and underlie coordinate descriptions: R2=R×R\mathbb{R}^2=\mathbb{R}\times\mathbb{R}R2=R×R, graphs of functions are subsets of X×YX\times YX×Y, and relations are subsets of products. Examples: If A={1,2}A=\{1,2\}A={1,2} and B={a,b}B=\{a,b\}B={a,b}, then A×B={(1,a),(1,b),(2,a),(2,b)}A\times B=\{(1,a),(1,b),(2,a),(2,b)\}A×B={(1,a),(1,b),(2,a),(2,b)}. R3=R×R×R\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}R3=R×R×R. If A=∅A=\varnothingA=∅, then A×B=∅A\times B=\varnothingA×B=∅ for any set BBB.