Let (X,d)(X,d)(X,d) be a metric space and let E⊆XE\subseteq XE⊆X. The set EEE is path-connected if for every x,y∈Ex,y\in Ex,y∈E there exists a path γ:[0,1]→E\gamma:[0,1]\to Eγ:[0,1]→E such that γ(0)=xandγ(1)=y.\gamma(0)=x\quad\text{and}\quad \gamma(1)=y.γ(0)=xandγ(1)=y.Path-connectedness implies connectedness in metric spaces (and more generally in topological spaces). In Euclidean spaces, many natural sets are path-connected by explicit paths. Examples: Any interval in R\mathbb{R}R is path-connected via linear interpolation. Any convex subset C⊆RkC\subseteq\mathbb{R}^kC⊆Rk is path-connected: use γ(t)=(1−t)x+ty\gamma(t)=(1-t)x+tyγ(t)=(1−t)x+ty. The set R2∖{0}\mathbb{R}^2\setminus\{0\}R2∖{0} is path-connected (e.g., connect points by polygonal paths avoiding 000).

Let XXX be a set, let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space , and let F\mathcal{F}F be a family of functions f:X→Yf:X\to Yf:X→Y. The family F\mathcal{F}F is pointwise bounded if for every x∈Xx\in Xx∈X the set of values F(x)={f(x):f∈F}⊆Y \mathcal{F}(x)=\{f(x): f\in\mathcal{F}\}\subseteq Y F(x)={f(x):f∈F}⊆Y is bounded in YYY, meaning there exist yx∈Yy_x\in Yyx​∈Y and Mx<∞M_x<\inftyMx​<∞ such that dY(f(x),yx)≤Mxfor all f∈F. d_Y\bigl(f(x),y_x\bigr)\le M_x\quad\text{for all } f\in\mathcal{F}. dY​(f(x),yx​)≤Mx​for all f∈F. ...

Let XXX be a set and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space . A sequence of functions fn:X→Yf_n:X\to Yfn​:X→Y converges pointwise to a function f:X→Yf:X\to Yf:X→Y if ∀x∈X,lim⁡n→∞dY(fn(x),f(x))=0.\forall x\in X,\quad \lim_{n\to\infty} d_Y\bigl(f_n(x),f(x)\bigr)=0.∀x∈X,n→∞lim​dY​(fn​(x),f(x))=0. One writes fn(x)→f(x)f_n(x)\to f(x)fn​(x)→f(x) for each fixed xxx. ...

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous on [a,b][a,b][a,b] and differentiable on (a,b)(a,b)(a,b). Corollary: If f′(x)>0f'(x)>0f′(x)>0 for all x∈(a,b)x\in(a,b)x∈(a,b), then fff is strictly increasing on [a,b][a,b][a,b]. Connection to parent theorem: Apply the mean value theorem to any x<yx<yx<y in [a,b][a,b][a,b] to get f(y)−f(x)=f′(c)(y−x)f(y)-f(x)=f'(c)(y-x)f(y)−f(x)=f′(c)(y−x) for some c∈(x,y)c\in(x,y)c∈(x,y). Since f′(c)>0f'(c)>0f′(c)>0 and y−x>0y-x>0y−x>0, one has f(y)>f(x)f(y)>f(x)f(y)>f(x). ...

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 be a power series with radius of convergence R>0R>0R>0. Corollary: For every xxx with ∣x−x0∣<R|x-x_0|<R∣x−x0​∣<R, the function fff is C∞C^\inftyC∞ (infinitely differentiable ) and for each k≥1k\ge 1k≥1, f(k)(x)=∑n=k∞n(n−1)⋯(n−k+1) an(x−x0)n−k. f^{(k)}(x)=\sum_{n=k}^\infty n(n-1)\cdots(n-k+1)\,a_n (x-x_0)^{n-k}. f(k)(x)=∑n=k∞​n(n−1)⋯(n−k+1)an​(x−x0​)n−k. In particular, the Taylor series of fff at x0x_0x0​ is exactly the original power series: f(x)=∑n=0∞f(n)(x0)n!(x−x0)n(∣x−x0∣<R). f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \quad (|x-x_0|<R). f(x)=∑n=0∞​n!f(n)(x0​)​(x−x0​)n(∣x−x0​∣<R). ...

The power set of a set AAA is P(A):={B:B⊆A}.\mathcal{P}(A) := \{B : B\subseteq A\}.P(A):={B:B⊆A}.Power sets organize all possible subcollections of AAA. In analysis they occur implicitly whenever one studies families of subsets (e.g., open sets, measurable sets) as subsets of P(X)\mathcal{P}(X)P(X) for some ambient set XXX. Examples: If A={1,2}A=\{1,2\}A={1,2}, then P(A)={∅,{1},{2},{1,2}}\mathcal{P}(A)=\{\varnothing,\{1\},\{2\},\{1,2\}\}P(A)={∅,{1},{2},{1,2}}. P(∅)={∅}\mathcal{P}(\varnothing)=\{\varnothing\}P(∅)={∅}. If AAA is finite with ∣A∣=n|A|=n∣A∣=n, then ∣P(A)∣=2n|\mathcal{P}(A)|=2^n∣P(A)∣=2n.

Let f:X→Yf:X\to Yf:X→Y be a function and let B⊆YB\subseteq YB⊆Y. The preimage (or inverse image) of BBB under fff is f−1(B):={x∈X:f(x)∈B}⊆X.f^{-1}(B):=\{x\in X : f(x)\in B\}\subseteq X.f−1(B):={x∈X:f(x)∈B}⊆X.The notation f−1(B)f^{-1}(B)f−1(B) does not require fff to be invertible; it is defined for every function. Preimages interact well with set operations and are central in topology (continuity via preimages of open sets) and measure theory (measurability via preimages of measurable sets). ...

The principle of mathematical induction states: if P(n)P(n)P(n) is a statement depending on n∈Nn\in\mathbb{N}n∈N and P(1)P(1)P(1) is true (base case), and for every n∈Nn\in\mathbb{N}n∈N, P(n)⇒P(n+1)P(n)\Rightarrow P(n+1)P(n)⇒P(n+1) (inductive step), then P(n)P(n)P(n) is true for all n∈Nn\in\mathbb{N}n∈N. Induction underlies virtually every rigorous argument about integers, sequences, and series, and is used to prove inequalities, formulas for sums, and structural properties of N\mathbb{N}N. ...

A proper subset of a set BBB is a subset AAA of BBB that is not equal to BBB. Formally, A⊊Bmeans(A⊆B) ∧ (A≠B).A \subsetneq B \quad\text{means}\quad (A\subseteq B)\ \land\ (A\neq B).A⊊Bmeans(A⊆B) ∧ (A=B).Proper inclusion ⊊\subsetneq⊊ indicates strict containment. In analysis it is used, for example, to describe dense proper subsets (e.g., Q⊊R\mathbb{Q}\subsetneq\mathbb{R}Q⊊R) and strict refinements of covers or partitions. ...

Ratio Test: Let ∑an\sum a_n∑an​ be a real or complex series with an≠0a_n\neq 0an​=0 eventually. Define ρ=lim sup⁡n→∞∣an+1an∣.\rho=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.ρ=limsupn→∞​​an​an+1​​​. ...