Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces . On the product X×YX\times YX×Y, define the metric d∞((x,y),(x′,y′))=max⁡{dX(x,x′), dY(y,y′)}. d_\infty\bigl((x,y),(x',y')\bigr)=\max\{d_X(x,x'),\,d_Y(y,y')\}. d∞​((x,y),(x′,y′))=max{dX​(x,x′),dY​(y,y′)}. (Any equivalent product metric, such as d1=dX+dYd_1=d_X+d_Yd1​=dX​+dY​, yields the same notion of convergence .) ...

Convergent implies Cauchy: Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn​) be a sequence in XXX. If xn→xx_n\to xxn​→x for some x∈Xx\in Xx∈X, then (xn)(x_n)(xn​) is a Cauchy sequence : ∀ε>0 ∃N ∀m,n≥N: d(xn,xm)<ε. \forall\varepsilon>0\;\exists N\;\forall m,n\ge N:\ d(x_n,x_m)<\varepsilon. ∀ε>0∃N∀m,n≥N: d(xn​,xm​)<ε. ...

Let (X,d)(X,d)(X,d) be a metric space and let (xn)n∈N(x_n)_{n\in\mathbb{N}}(xn​)n∈N​ be a sequence in XXX. The sequence is convergent if there exists x∈Xx\in Xx∈X such that ∀ε>0, ∃N∈N such that ∀n≥N, d(xn,x)<ε.\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n\ge N,\ d(x_n,x)<\varepsilon.∀ε>0, ∃N∈N such that ∀n≥N, d(xn​,x)<ε. In that case, one writes xn→xx_n\to xxn​→x and calls xxx the limit . ...

A series ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ (with an∈Ra_n\in\mathbb{R}an​∈R or C\mathbb{C}C) is convergent if its partial sums sN:=∑n=1Nans_N := \sum_{n=1}^N a_nsN​:=n=1∑N​an​ converge to a limit sss as N→∞N\to\inftyN→∞. In that case one writes ...

Corollary: If ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ converges in R\mathbb{R}R or C\mathbb{C}C, then an→0as n→∞. a_n\to 0 \quad\text{as } n\to\infty. an​→0as n→∞. This is a necessary condition for convergence of a series (but far from sufficient). Connection to parent theorem: Let sN=∑n=1Nans_N=\sum_{n=1}^N a_nsN​=∑n=1N​an​ be the partial sums . If sN→ss_N\to ssN​→s, then aN=sN−sN−1→s−s=0. a_N = s_N - s_{N-1}\to s-s=0. aN​=sN​−sN−1​→s−s=0. ...

Let f:I→Rf:I\to\mathbb{R}f:I→R be defined on an interval I⊆RI\subseteq\mathbb{R}I⊆R, and let a∈Ia\in Ia∈I. The point aaa is a critical point of fff if either: f′(a)=0f'(a)=0f′(a)=0, or f′(a)f'(a)f′(a) does not exist. Critical points are where local extrema can occur (Fermat’s principle: interior local extrema force f′(a)=0f'(a)=0f′(a)=0 when differentiable ). They are the main inputs for optimization via derivative tests. ...

A (parametrized) curve in Rk\mathbb{R}^kRk is a function γ:[a,b]→Rk,\gamma:[a,b]\to \mathbb{R}^k,γ:[a,b]→Rk, usually assumed at least continuous (and often differentiable or C1C^1C1 depending on context). The set γ([a,b])⊆Rk\gamma([a,b])\subseteq\mathbb{R}^kγ([a,b])⊆Rk is the trace (or image) of the curve. Curves provide a way to describe paths, boundaries, and parameterized geometry. In analysis, curves are used to build paths in spaces and to study continuous images of intervals. ...

Darboux’s Theorem: Let f:(a,b)→Rf:(a,b)\to\mathbb{R}f:(a,b)→R be differentiable . If x1,x2∈(a,b)x_1,x_2\in(a,b)x1​,x2​∈(a,b) with x1<x2x_1<x_2x1​<x2​ and α\alphaα lies between f′(x1)f'(x_1)f′(x1​) and f′(x2)f'(x_2)f′(x2​), then there exists c∈(x1,x2)c\in(x_1,x_2)c∈(x1​,x2​) such that f′(c)=α. f'(c)=\alpha. f′(c)=α. ...

Let (X,d)(X,d)(X,d) be a metric space and let D⊆XD\subseteq XD⊆X. The set DDD is dense in XXX if its closure equals XXX: D‾=X, \overline{D}=X, D=X, where D‾\overline{D}D is the intersection of all closed subsets of XXX that contain DDD. Equivalently, DDD is dense in XXX if and only if any (hence all) of the following hold: ...

Let (X,d)(X,d)(X,d) be a metric space and let D⊆XD\subseteq XD⊆X. The set DDD is dense in XXX if D‾=X\overline{D}=XD=X (see closure ). Equivalently, DDD is dense in XXX iff for every nonempty open set U⊆XU\subseteq XU⊆X, one has U∩D≠∅U\cap D\neq\varnothingU∩D=∅. Density means that every point of XXX can be approximated arbitrarily well by points of DDD. Dense subsets are central in approximation theorems and in separability questions. ...