Term-by-term differentiation of power series: Let f(x)=∑n=0∞an(x−x0)n f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n f(x)=∑n=0∞​an​(x−x0​)n have radius of convergence R>0R>0R>0. Define the derived series ∑n=1∞nan(x−x0)n−1. \sum_{n=1}^\infty n a_n (x-x_0)^{n-1}. ∑n=1∞​nan​(x−x0​)n−1. Then: ...

Term-by-term integration of power series: Let f(x)=∑n=0∞an(x−x0)n f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n f(x)=∑n=0∞​an​(x−x0​)n have radius of convergence R>0R>0R>0. Define F(x)=∑n=0∞ann+1(x−x0)n+1. F(x)=\sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1}. F(x)=∑n=0∞​n+1an​​(x−x0​)n+1. Then FFF has the same radius of convergence RRR, and for all ∣x−x0∣<R|x-x_0|<R∣x−x0​∣<R, F′(x)=f(x). F'(x)=f(x). F′(x)=f(x). Equivalently, for xxx with ∣x−x0∣<R|x-x_0|<R∣x−x0​∣<R, ∫x0xf(t) dt=∑n=0∞ann+1(x−x0)n+1. \int_{x_0}^{x} f(t)\,dt = \sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1}. ∫x0​x​f(t)dt=∑n=0∞​n+1an​​(x−x0​)n+1. ...

Let ∑n=1∞fn\sum_{n=1}^\infty f_n∑n=1∞​fn​ be a series of functions on an interval [a,b][a,b][a,b]. Proposition (term-by-term integration): Suppose each fnf_nfn​ is Riemann integrable on [a,b][a,b][a,b] and ∑fn\sum f_n∑fn​ converges uniformly on [a,b][a,b][a,b] to a function fff. Then fff is Riemann integrable and ∫abf(x) dx=∑n=1∞∫abfn(x) dx. \int_a^b f(x)\,dx=\sum_{n=1}^\infty \int_a^b f_n(x)\,dx. ∫ab​f(x)dx=∑n=1∞​∫ab​fn​(x)dx. ...

Let (X,d)(X,d)(X,d) be a metric space and let E⊆XE\subseteq XE⊆X. An ε\varepsilonε-net for EEE is a finite set {x1,…,xN}⊆X\{x_1,\dots,x_N\}\subseteq X{x1​,…,xN​}⊆X such that E⊆⋃j=1NHAHAHUGOSHORTCODE810s1HBHB(xj,ε). E\subseteq \bigcup_{j=1}^N B (x_j,\varepsilon). E⊆⋃j=1N​HAHAHUGOSHORTCODE810s1HBHB(xj​,ε). ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm. The function fff is (Fréchet) differentiable at a∈Ua\in Ua∈U if there exists a linear map A:Rk→RmA:\mathbb{R}^k\to\mathbb{R}^mA:Rk→Rm such that lim⁡h→0∥f(a+h)−f(a)−Ah∥2∥h∥2=0. \lim_{h\to 0}\frac{\|f(a+h)-f(a)-A h\|_2}{\|h\|_2}=0. h→0lim​∥h∥2​∥f(a+h)−f(a)−Ah∥2​​=0. The map AAA is uniquely determined when it exists and is called the (total) derivative of fff at aaa, denoted Df(a)Df(a)Df(a). ...

A total order (or linear order) on a set XXX is a partial order ≤\le≤ on XXX such that for all x,y∈Xx,y\in Xx,y∈X, x≤y or y≤x.x\le y\ \text{or}\ y\le x.x≤y or y≤x. This property is called comparability (or trichotomy when strengthened appropriately). Total orders allow one to speak of intervals, monotonicity, and order convergence, which are central in one-variable real analysis. ...

Let (X,d)(X,d)(X,d) be a metric space and let E⊆XE\subseteq XE⊆X. The set EEE is totally bounded if for every ε>0\varepsilon>0ε>0 there exist points x1,…,xN∈Xx_1,\dots,x_N\in Xx1​,…,xN​∈X such that E⊆⋃j=1NHAHAHUGOSHORTCODE818s2HBHB(xj,ε). E\subseteq \bigcup_{j=1}^N $B$ (x_j,\varepsilon). E⊆⋃j=1N​HAHAHUGOSHORTCODE818s2HBHB(xj​,ε). ...

Let (X,d)(X,d)(X,d) be a metric space and let E⊆XE\subseteq XE⊆X. The set EEE is totally bounded if for every ε>0\varepsilon>0ε>0 there exist points x1,…,xN∈Xx_1,\dots,x_N\in Xx1​,…,xN​∈X such that E⊆⋃j=1NB(xj,ε)E \subseteq \bigcup_{j=1}^N B(x_j,\varepsilon)E⊆j=1⋃N​B(xj​,ε) (see open ball ). ...

Triangle inequality (metric form): In a metric space (X,d)(X,d)(X,d), for all x,y,z∈Xx,y,z\in Xx,y,z∈X, d(x,z)≤d(x,y)+d(y,z). d(x,z)\le d(x,y)+d(y,z). d(x,z)≤d(x,y)+d(y,z). Triangle inequality (norm form): In a normed vector space (V,∥⋅∥)(V,\|\cdot\|)(V,∥⋅∥), for all u,v∈Vu,v\in Vu,v∈V, ∥u+v∥≤∥u∥+∥v∥. \|u+v\|\le \|u\|+\|v\|. ∥u+v∥≤∥u∥+∥v∥. The triangle inequality is the foundational estimate behind most ε\varepsilonε–arguments in analysis, including limit uniqueness, Cauchy criteria , and continuity estimates. ...

Uniform Cauchy implies uniform convergence: Let XXX be a set and let (Y,d)(Y,d)(Y,d) be a complete metric space . If (fn)(f_n)(fn​) is a uniformly Cauchy sequence of functions fn:X→Yf_n:X\to Yfn​:X→Y, then there exists a function f:X→Yf:X\to Yf:X→Y such that fn→ff_n\to ffn​→f uniformly on XXX. ...