The real numbers R\mathbb{R}R are characterized (up to a unique order-preserving field isomorphism) by the following structure: (R,+,⋅)(\mathbb{R},+,\cdot)(R,+,⋅) is a field: (R,+)(\mathbb{R},+)(R,+) is an abelian group with identity 000, (R∖{0},⋅)(\mathbb{R}\setminus\{0\},\cdot)(R∖{0},⋅) is an abelian group with identity 111, and multiplication distributes over addition. ≤\le≤ is a total order on R\mathbb{R}R compatible with the field operations, meaning: if a≤ba\le ba≤b then a+c≤b+ca+c\le b+ca+c≤b+c for all c∈Rc\in\mathbb{R}c∈R, if 0≤a0\le a0≤a and 0≤b0\le b0≤b then 0≤ab0\le ab0≤ab. (Completeness / least upper bound property) Every nonempty subset S⊆RS\subseteq\mathbb{R}S⊆R that is bounded above has a supremum in R\mathbb{R}R. ...

Let ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ be a series in R\mathbb{R}R or C\mathbb{C}C. A rearrangement of the series is any series of the form ∑n=1∞aσ(n),\sum_{n=1}^\infty a_{\sigma(n)},n=1∑∞​aσ(n)​, where σ:N→N\sigma:\mathbb{N}\to\mathbb{N}σ:N→N is a bijection (a permutation of the indices). ...

Rearrangement theorem (absolute convergence): If ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ converges absolutely in R\mathbb{R}R or C\mathbb{C}C and π:N→N\pi:\mathbb{N}\to\mathbb{N}π:N→N is a bijection, then the rearranged series ∑n=1∞aπ(n)\sum_{n=1}^\infty a_{\pi(n)}∑n=1∞​aπ(n)​ converges, and ∑n=1∞aπ(n)=∑n=1∞an.\sum_{n=1}^\infty a_{\pi(n)}=\sum_{n=1}^\infty a_n.∑n=1∞​aπ(n)​=∑n=1∞​an​. ...

Refinement lemma: Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be bounded . If PPP and QQQ are partitions of [a,b][a,b][a,b] and QQQ is a refinement of PPP (i.e., every partition point of PPP is also a partition point of QQQ), then L(f,P)≤L(f,Q)≤U(f,Q)≤U(f,P), L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P), L(f,P)≤L(f,Q)≤U(f,Q)≤U(f,P), where L(f,P)L(f,P)L(f,P) and U(f,P)U(f,P)U(f,P) are the lower and upper Riemann sums of fff with respect to PPP. ...

Let PPP and QQQ be partitions of [a,b][a,b][a,b]. The partition QQQ is a refinement of PPP if every point of PPP is also a point of QQQ, i.e. {x0,…,xn}⊆{y0,…,ym},\{x_0,\dots,x_n\}\subseteq \{y_0,\dots,y_m\},{x0​,…,xn​}⊆{y0​,…,ym​}, where P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0​<⋯<xn​=b and Q:a=y0<⋯<ym=bQ:a=y_0<\cdots<y_m=bQ:a=y0​<⋯<ym​=b. ...

Let U⊆RnU\subseteq \mathbb{R}^nU⊆Rn be open and let f:U→Rmf:U\to \mathbb{R}^mf:U→Rm be differentiable . A point a∈Ua\in Ua∈U is a regular point of fff if the derivative (Jacobian ) Df(a):Rn→RmDf(a):\mathbb{R}^n\to\mathbb{R}^mDf(a):Rn→Rm has rank mmm (equivalently, Df(a)Df(a)Df(a) is surjective). A point a∈Ua\in Ua∈U is a critical point of fff if rank⁡Df(a)<m\operatorname{rank} Df(a) < mrankDf(a)<m. ...

Let U⊆RnU\subseteq \mathbb{R}^nU⊆Rn be open and let f:U→Rmf:U\to \mathbb{R}^mf:U→Rm be differentiable . A value b∈Rmb\in \mathbb{R}^mb∈Rm is a regular value of fff if for every a∈f−1({b})a\in f^{-1}(\{b\})a∈f−1({b}) the point aaa is a regular point , i.e. ∀a∈U, f(a)=b ⟹ rank⁡Df(a)=m.\forall a\in U,\; f(a)=b \implies \operatorname{rank} Df(a)=m.∀a∈U,f(a)=b⟹rankDf(a)=m. A value bbb is a critical value if it is not a regular value; equivalently, there exists a∈Ua\in Ua∈U such that f(a)=bf(a)=bf(a)=b and rank⁡Df(a)<m\operatorname{rank}Df(a)<mrankDf(a)<m. ...

Let XXX and YYY be sets. A relation from XXX to YYY is a subset R⊆X×Y.R \subseteq X\times Y.R⊆X×Y. If (x,y)∈R(x,y)\in R(x,y)∈R, one often writes xRyxRyxRy. A relation on XXX is a subset R⊆X×XR\subseteq X\times XR⊆X×X. Relations generalize functions by allowing an input xxx to be related to zero, one, or many outputs yyy. Equivalence relations and order relations are special kinds of relations that structure sets. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The set AAA is relatively compact (or precompact) in XXX if its closure A‾\overline{A}A is compact in XXX: A‾ is compact. \overline{A}\ \text{is compact}. A is compact. Equivalently (in metric spaces), AAA is relatively compact if and only if every sequence in AAA has a convergent subsequence in XXX whose limit lies in A‾\overline{A}A. ...

Let fff be a function for which the Taylor polynomial Tkf(x;a)T_k f(x;a)Tk​f(x;a) is defined. The remainder term of order kkk (about aaa) is the function Rk(x;a):=f(x)−Tkf(x;a).R_k(x;a):= f(x)-T_k f(x;a).Rk​(x;a):=f(x)−Tk​f(x;a).Taylor’s theorem gives hypotheses under which Rk(x;a)R_k(x;a)Rk​(x;a) can be bounded or represented explicitly (e.g., Lagrange form or integral form), making the approximation f(x)≈Tkf(x;a)f(x)\approx T_k f(x;a)f(x)≈Tk​f(x;a) quantitatively precise. ...